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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre .
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how are you supposed to get the triangles the right size ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes .
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is there any relationship between a hexaflexagon and a mobius strip ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes .
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could someone please explain what a mobius strip is and how it relates to a hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa !
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is there a possibility for a 20 sided hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class .
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how do the hexaflexagons have more than two sides ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird .
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is a hexaflexagon like a rubix cube in any way ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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can u make a movie on how to make these hexaflexagons ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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can someone give good instructions to make a hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side .
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is it possible to get more sides than a hexaflexagon ( more than six sides ) ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper .
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how old was arthur h.stone when he discovered the hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow .
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can there be another shape you can make from the mobius strip that you can flex ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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can someone please tell me how to make a hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out .
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have you noticed that vi 's sleeves and nail varnish are different colors ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out .
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how was the 2nd hexaflexagone flipping more than the 1st ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips .
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how long did it take for people to come up with the different types of hexaflexagons ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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does anybody know any websites that can teach you how to make a hexaflexagon slowly or in written form ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa !
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are there groups in hexaflexagon or are all the states always available ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink .
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what do these hexaflexagons and mobius strips have to do with math ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong .
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how do you make equilateral triangles in the strip without having to use a compass ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper .
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how wide/long does the strip of paper have to be ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out .
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do vi 's video articles lead anyone to recall author martin gardner ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out .
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can someone give me vi hart 's website ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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is there a way to make an octagon instead of a hexagon to make a octaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side .
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if the flexagons are indeed mobius strips , if i make a mobius strip and collapse it will i get a flexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper .
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like , for example , did the paper strips actually come from leftover binder paper ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how do you make a flexagon without a pattern ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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is it possible to make hexaflexagons with over 3 colors ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class .
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also how do you color the hexaflexagon before you put it together so you do n't have to find all of the sides ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . ''
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how does vi hart just take one thing like a hexagon and turn it into something amazing and awesome ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa !
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does anyone know why certain faces of a hexaflexagon occur more often that others ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa !
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what is the approximate height and width of a hexaflexagon in inches ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how the heck do you make a flexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side .
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how does the one have so many different sides ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side .
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i wonder how many sides you can make in a flexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how do you make equilateral triangles so that a hexaflexagon can be made ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon .
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would n't you get in trouble if you mess around with the paper strips during math class ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out .
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is it just that vi is a genius ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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why is is hard to make flexagon out of aluminum foil ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper .
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why do you say to use a strip of 9 triangles for tri hexaflexagon and a strip of 18 for a hexa hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side .
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how would you be able to make a flexagon that is triangle shaped ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is .
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how many triangles do you need for the trihexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes .
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what is the difference between a hexaflexagon and a mobius strip ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy !
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how do you make one of the hexaflexagons ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how do you make an equilateral triangle ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps .
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is there a way to make a hexaflexagon with even more sides than 6 ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons .
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why does the hexahexaflexagon `` flex '' more than the hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out .
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how do you talk so fast , vi ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how do you make the strip ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper .
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who is arthur h stone is 1939 ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how do you make the hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes .
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what is a mobius strip ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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i cant make a hexflexagon can someone tell me how ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this .
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can you make a `` flexagon '' using other shapes besides hexagons ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how do you make the hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink .
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if you drew a circle in the middle of the hexaflexagon , and you flexed it , would the circle be on the edges ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green .
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did any one notice that vi 's sleeves are 2 different colors ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink .
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how is a piece of paper supposed to be folded into a piece of paper ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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can you make a slower version of how to make a hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange .
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how do we actually fold a hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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could someone please explain to me how to make a hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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orange , yellow , white ? wait a second . okay , so you colour that one green .
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how many colors can you do on the second hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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can anybody teach me how to make a 3d hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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vi , can you please tell me how to make a hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how to make a hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa !
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does a bigger hexaflexagon have more colors ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa !
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is there a specific relation between spheres hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink .
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i guess that it would be possible to create hexaflexagons with an infinite amount of faces if you could get paper that was thin enough ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps .
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would there be any way to turn a series of hexaflexagons into a 3-d shape ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . ''
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like how you can turn mobius strips into klein bottles ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper .
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how do you make a paper hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy !
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what are the instuctions to make one ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon .
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how can i get longer paper strips ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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orange , yellow , white ? wait a second . okay , so you colour that one green .
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wait wiat wait wait ... what happend to being vi in math class ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre .
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why are my triangles never correct ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah .
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is there any class codes or teacher 's emails available for me to have ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper .
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so , if this hexaflexagon is a true story , did arthur ever do anything else other than hexaflexagons ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how does she make the paper fold so that you can make a 4 sided hexiflexigon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa !
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what are the dimensions for the trihexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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53 , how do you make that 6-flapped hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink .
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how do the several sides actually work ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ?
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why is the way to make a hexaflexagon counted 4-4-5-5-6-6-4-4-5-5-6-6-4-4-5-5-6-6 and not 4-5-6-4-5-6-4-5-6-4-5-6-4-5-6-4-5-6 on the back side ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa !
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has anyone else gotten stuck in an infinite loop with the hexa-hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side .
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how do you make it so that there are more sides with different colors on it so its not a tri-hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green .
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can someone give me their input on whether a trihexaflexagon or a regular one is better ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre .
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because you need 9 equilateral triangles ( ) , is that why there are 3 `` sides '' and a hexagon only has 6 triangles ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper .
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does it matter how wide my strip of paper is ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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can you please show me how to make a hexaflexagon in slow motion ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes .
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what are the differences between a hexaflexagon and a mobius strip ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . ''
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why does vi hart make herself look like she 's about to eat hexaflexagons ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons .
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how do you flex the 6-sided hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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can you make a video on how to make a 4-sided hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this .
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besides the mobius loop ( at least i think its a mobius loop ) , in the first hexaflexagons video , would anyone be willing to share the names and/or folding instructions for the other shapes that vi creates from the strips of paper starting ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons .
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how does the hexaflexagon flex without breaking ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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if vi fast forwards the video to make it go faster , wo n't her voice become higher ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa !
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what happens if you unfold a colored hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper .
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why does arthur h. stone have yellow nail polish/paint ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper .
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wait , is arthur h. stone real ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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is a hexaflexagon ezey to make ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . ''
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if a mobius strip is like a 180 turn , and a loop is a 360 turn , is n't a hexaflexagon like 2 mobius strips , considering it has a tri-fold on it 's loop ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong .
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what happens if you double-fold the strip of paper before maki a hexflexagon ?
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