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9,900 | On January 16, 2013, All Nippon Airways Flight NH-692, en route from Yamaguchi Ube Airport to Tokyo Haneda, had a battery problem warning followed by a burning smell while climbing from Ube about west of Takamatsu, Japan. The aircraft diverted to Takamatsu and was evacuated via the slides; three passengers received minor injuries during the evacuation. Inspection revealed a battery fire. A similar incident in a parked Japan Airlines 787 at Boston's Logan International Airport within the same week led the Federal Aviation Administration to ground all 787s. On January 16, 2013, both major Japanese airlines ANA and JAL voluntarily grounded their fleets of 787s after multiple incidents involving different 787s, including emergency landings. At the time, these two carriers operated 24 of the 50 787s delivered. The grounding reportedly cost ANA some 9 billion yen (US$93 million) in lost sales. | https://en.wikipedia.org/wiki?curid=307133 |
9,901 | On January 16, 2013, the FAA issued an emergency airworthiness directive ordering all American-based airlines to ground their Boeing 787s until yet-to-be-determined modifications were made to the electrical system to reduce the risk of the battery overheating or catching fire. This was the first time that the FAA had grounded an airliner type since 1979. Industry experts disagreed on consequences of the grounding: Airbus was confident that Boeing would resolve the issue and that no airlines will switch plane type, while other experts saw the problem as "costly" and "could take upwards of a year". | https://en.wikipedia.org/wiki?curid=307133 |
9,902 | The FAA also conducted an extensive review of the 787's critical systems. The focus of the review was on the safety of the lithium-ion batteries made of lithium cobalt oxide (LiCoO). The 787 battery contract was signed in 2005, when this was the only type of lithium aerospace battery available, but since then newer and safer types (such as LiFePO), which provide less reaction energy with virtually no cobalt content to avoid cobalt's thermal runaway characteristic, have become available. FAA approved a 787 battery in 2007 with nine "special conditions". A battery approved by FAA (through Mobile Power Solutions) was made by Rose Electronics using Kokam cells; the batteries installed in the 787 are made by Yuasa. | https://en.wikipedia.org/wiki?curid=307133 |
9,903 | On January 20, the NTSB declared that overvoltage was not the cause of the Boston incident, as voltage did not exceed the battery limit of 32 V, and the charging unit passed tests. The battery had signs of short circuiting and thermal runaway. Despite this, by January 24, the NTSB had not yet pinpointed the cause of the Boston fire; the FAA would not allow U.S.-based 787s to fly again until the problem was found and corrected. In a press briefing that day, NTSB Chairwoman Deborah Hersman said that the NTSB had found evidence of failure of multiple safety systems designed to prevent these battery problems, and stated that fire must never happen on an airplane. | https://en.wikipedia.org/wiki?curid=307133 |
9,904 | The Japan Transport Safety Board (JTSB) has said on January 23 that the battery in ANA jets in Japan reached a maximum voltage of 31 V (below the 32 V limit like the Boston JAL 787), but had a sudden unexplained voltage drop to near zero. All cells had signs of thermal damage prior to runaway. ANA and JAL had replaced several 787 batteries before the mishaps. , JTSB approved the Yuasa factory quality control while the NTSB examined the Boston battery for defects. The failure rate, with two major battery thermal runaway events in 100,000 flight hours, was much higher than the rate of one in 10 million flight hours predicted by Boeing. | https://en.wikipedia.org/wiki?curid=307133 |
9,905 | The only American airline that operated the Dreamliner at the time was United Airlines, which had six. Chile's Directorate General of Civil Aviation (DGAC) grounded LAN Airlines' three 787s. The Indian Directorate General of Civil Aviation (DGCA) directed Air India to ground its six Dreamliners. The Japanese Transport Ministry made the ANA and JAL groundings official and indefinite following the FAA announcement. The European Aviation Safety Agency also followed the FAA's advice and grounded the only two European 787s operated by LOT Polish Airlines. Qatar Airways grounded their five Dreamliners. Ethiopian Airlines was the final operator to temporarily ground its four Dreamliners. By January 17, 2013, all 50 of the aircraft delivered to date had been grounded. On January 18, Boeing halted 787 deliveries until the battery problem was resolved. | https://en.wikipedia.org/wiki?curid=307133 |
9,906 | On February 7, 2013, the FAA gave approval for Boeing to conduct 787 test flights to gather additional data. In February 2013, FAA oversight of the 787's 2007 safety approval and certification was under scrutiny. On March 7, 2013, the NTSB released an interim factual report about the Boston battery fire on January 7, 2013. The investigation stated that "heavy smoke and fire coming from the front of the APU battery case." Firefighters "tried fire extinguishing, but smoke and flame (flame size about 3 inches) did not stop". | https://en.wikipedia.org/wiki?curid=307133 |
9,907 | Boeing completed its final tests on a revised battery design on April 5, 2013. The FAA approved Boeing's revised battery design with three additional, overlapping protection methods on April 19, 2013. The FAA published a directive on April 25 to provide instructions for retrofitting battery hardware before the 787s could return to flight. The repairs were expected to be completed in weeks. Following the FAA approval in the U.S. effective April 26, Japan approved resumption of Boeing 787 flights in the country on April 26, 2013. On April 27, 2013, Ethiopian Airlines took a 787 on the model's first commercial flight after battery system modifications. | https://en.wikipedia.org/wiki?curid=307133 |
9,908 | On January 14, 2014, a battery in a JAL 787 emitted smoke from the battery's protection exhaust while the aircraft was undergoing pre-flight maintenance at Tokyo Narita Airport. The battery partially melted in the incident; one of its eight lithium-ion cells had its relief port vent and fluid sprayed inside the battery's container. It was later reported that the battery may have reached a temperature as high as , and that Boeing did not understand the root cause of the failure. | https://en.wikipedia.org/wiki?curid=307133 |
9,909 | The NTSB has criticized FAA, Boeing, and battery manufacturers for the faults in a 2014 report. It also criticized the GE-made flight data and cockpit voice recorder in the same report. The enclosure Boeing added is heavier, negating the lighter battery potential. | https://en.wikipedia.org/wiki?curid=307133 |
9,910 | In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities formula_2 and formula_3 with formula_4, | https://en.wikipedia.org/wiki?curid=12386 |
9,911 | where the Greek letter phi ( or formula_5) denotes the golden ratio. The constant formula_6 satisfies the quadratic equation formula_7 and is an irrational number with a value of | https://en.wikipedia.org/wiki?curid=12386 |
9,912 | The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names. | https://en.wikipedia.org/wiki?curid=12386 |
9,913 | Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of formula_6—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation. | https://en.wikipedia.org/wiki?curid=12386 |
9,914 | Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle. | https://en.wikipedia.org/wiki?curid=12386 |
9,915 | One method for finding formula_6's closed form starts with the left fraction. Simplifying the fraction and substituting the reciprocal formula_13, | https://en.wikipedia.org/wiki?curid=12386 |
9,916 | Because formula_6 is a ratio between positive quantities, formula_6 is necessarily the positive root. The negative root is in fact the negative inverse formula_17, which shares many properties with the golden ratio. | https://en.wikipedia.org/wiki?curid=12386 |
9,917 | Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (an irrational number), surprising Pythagoreans. Euclid's "Elements" () provides several propositions and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows: | https://en.wikipedia.org/wiki?curid=12386 |
9,918 | The golden ratio was studied peripherally over the next millennium. Abu Kamil (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers. | https://en.wikipedia.org/wiki?curid=12386 |
9,919 | Luca Pacioli named his book "Divina proportione" (1509) after the ratio; the book, largely plagiarized from Piero della Francesca, explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the "sectio aurea" ('golden section'). Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio. | https://en.wikipedia.org/wiki?curid=12386 |
9,920 | German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) golden ratio was stated as "about formula_18" in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student. The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these: | https://en.wikipedia.org/wiki?curid=12386 |
9,921 | 18th-century mathematicians Abraham de Moivre, Nicolaus I Bernoulli, and Leonhard Euler used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet, for whom it was named "Binet's formula". Martin Ohm first used the German term "goldener Schnitt" ('golden section') to describe the ratio in 1835. James Sully used the equivalent English term in 1875. | https://en.wikipedia.org/wiki?curid=12386 |
9,922 | By 1910, inventor Mark Barr began using the Greek letter Phi as a symbol for the golden ratio. It has also been represented by tau the first letter of the ancient Greek τομή ('cut' or 'section'). | https://en.wikipedia.org/wiki?curid=12386 |
9,923 | The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron/dodecahedron, and uses the golden ratio ubiquitously. Between 1973 and 1974, Roger Penrose developed Penrose tiling, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern. This gained in interest after Dan Shechtman's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterward explained through analogies to the Penrose tiling. | https://en.wikipedia.org/wiki?curid=12386 |
9,924 | If we call the whole formula_19 and the longer part formula_20 then the second statement above becomes | https://en.wikipedia.org/wiki?curid=12386 |
9,925 | To say that the golden ratio formula_6 is rational means that formula_6 is a fraction formula_23 where formula_19 and formula_25 are integers. We may take formula_23 to be in lowest terms and formula_19 and formula_25 to be positive. But if formula_23 is in lowest terms, then the equally valued formula_30 is in still lower terms. That is a contradiction that follows from the assumption that formula_6 is rational. | https://en.wikipedia.org/wiki?curid=12386 |
9,926 | Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If formula_32 is rational, then formula_33 is also rational, which is a contradiction if it is already known that the square root of all non-square natural numbers are irrational. | https://en.wikipedia.org/wiki?curid=12386 |
9,927 | The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial | https://en.wikipedia.org/wiki?curid=12386 |
9,928 | which has roots formula_36 and formula_37 As the root of a quadratic polynomial, the golden ratio is a constructible number. | https://en.wikipedia.org/wiki?curid=12386 |
9,929 | The absolute value of this quantity corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, formula_39). | https://en.wikipedia.org/wiki?curid=12386 |
9,930 | The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with formula_6: | https://en.wikipedia.org/wiki?curid=12386 |
9,931 | The sequence of powers of formula_6 contains these values formula_42 formula_43 formula_44 formula_45 more generally, | https://en.wikipedia.org/wiki?curid=12386 |
9,932 | As a result, one can easily decompose any power of formula_6 into a multiple of formula_6 and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of formula_6: | https://en.wikipedia.org/wiki?curid=12386 |
9,933 | The formula formula_51 can be expanded recursively to obtain a continued fraction for the golden ratio: | https://en.wikipedia.org/wiki?curid=12386 |
9,934 | The convergents of these continued fractions formula_52 formula_52 formula_54 formula_55 formula_56 formula_57 ... or formula_58 formula_59 formula_60 formula_61 formula_62 are ratios of successive Fibonacci numbers. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the Hurwitz inequality for Diophantine approximations, which states that for every irrational formula_63, there are infinitely many distinct fractions formula_64 such that, | https://en.wikipedia.org/wiki?curid=12386 |
9,935 | This means that the constant formula_66 cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such Lagrange numbers. | https://en.wikipedia.org/wiki?curid=12386 |
9,936 | Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence formula_69: | https://en.wikipedia.org/wiki?curid=12386 |
9,937 | The sequence of Lucas numbers (not to be confused with the generalized Lucas sequences, of which this is part) is like the Fibonacci sequence, in-which each term is the sum of the previous two, however instead starts with formula_70: | https://en.wikipedia.org/wiki?curid=12386 |
9,938 | Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers: | https://en.wikipedia.org/wiki?curid=12386 |
9,939 | In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates formula_6. | https://en.wikipedia.org/wiki?curid=12386 |
9,940 | These approximations are alternately lower and higher than formula_74 and converge to formula_6 as the Fibonacci and Lucas numbers increase. | https://en.wikipedia.org/wiki?curid=12386 |
9,941 | Combining both formulas above, one obtains a formula for formula_76 that involves both Fibonacci and Lucas numbers: | https://en.wikipedia.org/wiki?curid=12386 |
9,942 | Between Fibonacci and Lucas numbers one can deduce formula_77 which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five: | https://en.wikipedia.org/wiki?curid=12386 |
9,943 | Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by infinite summation: | https://en.wikipedia.org/wiki?curid=12386 |
9,944 | In particular, the powers of formula_6 themselves round to Lucas numbers (in order, except for the first two powers, formula_85 and formula_6, are in reverse order): | https://en.wikipedia.org/wiki?curid=12386 |
9,945 | Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of "third" consecutive Fibonacci numbers equals a Lucas number, that is formula_88; and, importantly, that formula_89. | https://en.wikipedia.org/wiki?curid=12386 |
9,946 | Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the golden spiral (which is a special form of a logarithmic spiral) using quarter-circles with radii from these sequences, differing only slightly from the "true" golden logarithmic spiral. "Fibonacci spiral" is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles. | https://en.wikipedia.org/wiki?curid=12386 |
9,947 | The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the pentagon, and extends to form part of the coordinates of the vertices of a regular dodecahedron, as well as those of a 5-cell. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes. | https://en.wikipedia.org/wiki?curid=12386 |
9,948 | Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length. | https://en.wikipedia.org/wiki?curid=12386 |
9,949 | Both of the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer one to the shorter one is the golden ratio. | https://en.wikipedia.org/wiki?curid=12386 |
9,950 | When two angles that make a full circle have measures in the golden ratio, the smaller is called the "golden angle", with measure formula_123 | https://en.wikipedia.org/wiki?curid=12386 |
9,951 | This angle occurs in patterns of plant growth as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them. | https://en.wikipedia.org/wiki?curid=12386 |
9,952 | Logarithmic spirals are self-similar spirals where distances covered per turn are in geometric progression. Importantly, isosceles golden triangles can be encased by a golden logarithmic spiral, such that successive turns of a spiral generate new golden triangles inside. This special case of logarithmic spirals is called the "golden spiral", and it exhibits continuous growth in golden ratio. That is, for every formula_125 turn, there is a growth factor of formula_6. As mentioned above, these "golden spirals" can be approximated by quarter-circles generated from Fibonacci and Lucas number-sized squares that are tiled together. In their exact form, they can be described by the polar equation with formula_127: | https://en.wikipedia.org/wiki?curid=12386 |
9,953 | Golden spirals can be symmetrically placed inside pentagons and pentagrams as well, such that fractal copies of the underlying geometry are reproduced at all scales. | https://en.wikipedia.org/wiki?curid=12386 |
9,954 | George Odom found a construction for formula_6 involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. | https://en.wikipedia.org/wiki?curid=12386 |
9,955 | The "Kepler triangle", named after Johannes Kepler, is the unique right triangle with sides in geometric progression: | https://en.wikipedia.org/wiki?curid=12386 |
9,956 | The Kepler triangle can also be understood as the right triangle formed by three squares whose areas are also in golden geometric progression formula_135. | https://en.wikipedia.org/wiki?curid=12386 |
9,957 | Fittingly, the Pythagorean means for formula_136 are precisely formula_137, formula_6, and formula_139. It is from these ratios that we are able to geometrically express the fundamental defining quadratic polynomial for formula_6 with the Pythagorean theorem; that is, formula_7. | https://en.wikipedia.org/wiki?curid=12386 |
9,958 | The inradius of an isosceles triangle is greatest when the triangle is composed of two mirror Kepler triangles, such that their bases lie on the same line. Also, the isosceles triangle of given perimeter with the largest possible semicircle is one from two mirrored Kepler triangles. | https://en.wikipedia.org/wiki?curid=12386 |
9,959 | A "golden triangle" is characterized as an isosceles formula_143 with the property that bisecting the angle formula_144 produces new acute and obtuse isosceles triangles formula_145 and formula_146 that are similar to the original, as well as in leg to base length ratios of formula_147 and formula_148, respectively. | https://en.wikipedia.org/wiki?curid=12386 |
9,960 | The acute isosceles triangle is sometimes called a "sublime triangle", and the ratio of its base to its equal-length sides is formula_6. Its apex angle formula_150 is equal to: | https://en.wikipedia.org/wiki?curid=12386 |
9,961 | Both base angles of the isosceles golden triangle equal formula_151 degrees each, since the sum of the angles of a triangle must equal formula_152 degrees. It is the only triangle to have its three angles in formula_153 ratio. A regular pentagram contains five acute sublime triangles, and a regular decagon contains ten, as each two vertices connected to the center form acute golden triangles. | https://en.wikipedia.org/wiki?curid=12386 |
9,962 | The obtuse isosceles triangle is sometimes called a "golden gnomon", and the ratio of its base to its other sides is the reciprocal of the golden ratio, formula_154. The measure of its apex angle formula_155 is: | https://en.wikipedia.org/wiki?curid=12386 |
9,963 | Its two base angles equal formula_156 each. It is the only triangle whose internal angles are in formula_157 ratio. Its base angles, being equal to formula_156, are the same measure as that of the acute golden triangle's apex angle. Five golden gnomons can be created from adjacent sides of a pentagon whose non-coincident vertices are joined by a diagonal of the pentagon. | https://en.wikipedia.org/wiki?curid=12386 |
9,964 | Appropriately, the ratio of the area of the obtuse golden gnomon to that of the acute sublime triangle is in formula_159 golden ratio. Bisecting a base angle inside a sublime triangle produces a golden gnomon, and another a sublime triangle. Bisecting the apex angle of a golden gnomon in formula_160 ratio produces two new golden triangles, too. Golden triangles that are decomposed further like this into pairs of isosceles and obtuse golden triangles are known as "Robinson triangles." | https://en.wikipedia.org/wiki?curid=12386 |
9,965 | The golden ratio proportions the adjacent side lengths of a "golden rectangle" in formula_159 ratio. Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in formula_6 ratio. They can be generated by "golden spirals", through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the icosahedron as well as in the dodecahedron (see section below for more detail). | https://en.wikipedia.org/wiki?curid=12386 |
9,966 | A "golden rhombus" is a rhombus whose diagonals are in proportion to the golden ratio, most commonly formula_159. For a rhombus of such proportions, its acute angle and obtuse angles are: | https://en.wikipedia.org/wiki?curid=12386 |
9,967 | The lengths of its short and long diagonals formula_164 and formula_165, in terms of side length formula_2 are: | https://en.wikipedia.org/wiki?curid=12386 |
9,968 | These geometric values can be calculated from their Cartesian coordinates, which also can be given using formulas involving formula_6. The coordinates of the dodecahedron are displayed on the figure above, while those of the icosahedron are the cyclic permutations of: | https://en.wikipedia.org/wiki?curid=12386 |
9,969 | Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings. In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. In all, the three golden rectangles contain formula_172 vertices of the icosahedron, or equivalently, intersect the centers of formula_172 of the dodecahedron's faces. | https://en.wikipedia.org/wiki?curid=12386 |
9,970 | A cube can be inscribed in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is formula_174 times that of the dodecahedron's. In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in formula_175 ratio. On the other hand, the octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's formula_172 vertices touch the formula_172 edges of an octahedron at points that divide its edges in golden ratio. | https://en.wikipedia.org/wiki?curid=12386 |
9,971 | Other polyhedra are related to the dodecahedron and icosahedron or their symmetries, and therefore have corresponding relations to the golden ratio. These include the compound of five cubes, compound of five octahedra, compound of five tetrahedra, the compound of ten tetrahedra, rhombic triacontahedron, icosidodecahedron, truncated icosahedron, truncated dodecahedron, and rhombicosidodecahedron, rhombic enneacontahedron, and Kepler-Poinsot polyhedra, and rhombic hexecontahedron. In four dimensions, the dodecahedron and icosahedron appear as faces of the 120-cell and 600-cell, which again have dimensions related to the golden ratio. | https://en.wikipedia.org/wiki?curid=12386 |
9,972 | The golden ratio's "decimal expansion" can be calculated via root-finding methods, such as Newton's method or Halley's method, on the equation formula_178 or on formula_179 (to compute formula_66 first). The time needed to compute formula_19 digits of the golden ratio using Newton's method is essentially formula_182, where formula_183 is the time complexity of multiplying two formula_19-digit numbers. This is considerably faster than known algorithms for formula_185 and formula_186. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers formula_187 and formula_188 each over formula_189 digits, yields over formula_190 significant digits of the golden ratio. The decimal expansion of the golden ratio formula_6 has been calculated to an accuracy of ten trillion digits. | https://en.wikipedia.org/wiki?curid=12386 |
9,973 | The golden ratio and inverse golden ratio formula_192 have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations formula_193 – this fact corresponds to the identity and the definition quadratic equation. | https://en.wikipedia.org/wiki?curid=12386 |
9,974 | Further, they are interchanged by the three maps formula_194 – they are reciprocals, symmetric about formula_195 and (projectively) symmetric about formula_196 More deeply, these maps form a subgroup of the modular group formula_197 isomorphic to the symmetric group on formula_198 letters, formula_199 corresponding to the stabilizer of the set formula_200 of formula_198 standard points on the projective line, and the symmetries correspond to the quotient map formula_202 – the subgroup formula_203 consisting of the identity and the formula_198-cycles, in cycle notation formula_205 fixes the two numbers, while the formula_206-cycles formula_207 interchange these, thus realizing the map. | https://en.wikipedia.org/wiki?curid=12386 |
9,975 | In the complex plane, the fifth roots of unity formula_208 (for an integer formula_209) satisfying formula_210 are the vertices of a pentagon. They do not form a ring of quadratic integers, however the sum of any fifth root of unity and its complex conjugate, formula_211 "is" a quadratic integer, an element of formula_212 Specifically, | https://en.wikipedia.org/wiki?curid=12386 |
9,976 | For the gamma function formula_214, the only solutions to the equation formula_215 are formula_216 and formula_217. | https://en.wikipedia.org/wiki?curid=12386 |
9,977 | When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed "phinary" or formula_6"-nary"), quadratic integers in the ring formula_219 – that is, numbers of the form formula_220 for formula_221 – have terminating representations, but rational fractions have non-terminating representations. | https://en.wikipedia.org/wiki?curid=12386 |
9,978 | The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is formula_222 | https://en.wikipedia.org/wiki?curid=12386 |
9,979 | where formula_224 and formula_225 in the continued fraction should be evaluated as formula_226. The function formula_227 is invariant under formula_228, a congruence subgroup of the modular group. Also for positive real numbers formula_229 and formula_230 then | https://en.wikipedia.org/wiki?curid=12386 |
9,980 | The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned." | https://en.wikipedia.org/wiki?curid=12386 |
9,981 | Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. | https://en.wikipedia.org/wiki?curid=12386 |
9,982 | In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles. | https://en.wikipedia.org/wiki?curid=12386 |
9,983 | Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house. | https://en.wikipedia.org/wiki?curid=12386 |
9,984 | Leonardo da Vinci's illustrations of polyhedra in Pacioli's "Divina proportione" have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his "Mona Lisa", for example, employs golden ratio proportions, is not supported by Leonardo's own writings. Similarly, although Leonardo's "Vitruvian Man" is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios. | https://en.wikipedia.org/wiki?curid=12386 |
9,985 | Salvador Dalí, influenced by the works of Matila Ghyka, explicitly used the golden ratio in his masterpiece, "The Sacrament of the Last Supper". The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition. | https://en.wikipedia.org/wiki?curid=12386 |
9,986 | A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is formula_232 with averages for individual artists ranging from formula_233 (Goya) to formula_234 (Bellini). On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and formula_235 proportions, and others with proportions like formula_236 formula_237 formula_238 and formula_239 | https://en.wikipedia.org/wiki?curid=12386 |
9,987 | There was a time when deviations from the truly beautiful page proportions formula_240 formula_241 and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter. | https://en.wikipedia.org/wiki?curid=12386 |
9,988 | According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions. | https://en.wikipedia.org/wiki?curid=12386 |
9,989 | The aspect ratio (width to height ratio) of the flag of Togo was intended to be the golden ratio, according to its designer. | https://en.wikipedia.org/wiki?curid=12386 |
9,990 | Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. French composer Erik Satie used the golden ratio in several of his pieces, including "Sonneries de la Rose+Croix". The golden ratio is also apparent in the organization of the sections in the music of Debussy's "Reflets dans l'eau (Reflections in Water)", from "Images" (1st series, 1905), in which "the sequence of keys is marked out by the intervals and and the main climax sits at the phi position". | https://en.wikipedia.org/wiki?curid=12386 |
9,991 | The musicologist Roy Howat has observed that the formal boundaries of Debussy's "La Mer" correspond exactly to the golden section. Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions. | https://en.wikipedia.org/wiki?curid=12386 |
9,992 | Music theorists including Hans Zender and Heinz Bohlen have experimented with the 833 cents scale, a musical scale based on using the golden ratio as its fundamental musical interval. When measured in cents, a logarithmic scale for musical intervals, the golden ratio is approximately 833.09 cents. | https://en.wikipedia.org/wiki?curid=12386 |
9,993 | Johannes Kepler wrote that "the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio". | https://en.wikipedia.org/wiki?curid=12386 |
9,994 | The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law. Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art". | https://en.wikipedia.org/wiki?curid=12386 |
9,995 | However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious. | https://en.wikipedia.org/wiki?curid=12386 |
9,996 | The quasi-one-dimensional Ising ferromagnet CoNbO (cobalt niobate) has 8 predicted excitation states (with E symmetry), that when probed with neutron scattering, showed its lowest two were in golden ratio. Specifically, these quantum phase transitions during spin excitation, which occur at near absolute zero temperature, showed pairs of kinks in its ordered-phase to spin-flips in its paramagnetic phase; revealing, just below its critical field, a spin dynamics with sharp modes at low energies approaching the golden mean. | https://en.wikipedia.org/wiki?curid=12386 |
9,997 | There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, "Thomson problem" or "Tammes problem"). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. formula_242 This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3. | https://en.wikipedia.org/wiki?curid=12386 |
9,998 | The Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu) has been analyzed by pyramidologists as having a doubled Kepler triangle as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on pi or on whole-number ratios. The consensus of modern scholars is that this pyramid's proportions are not based on the golden ratio, because such a basis would be inconsistent both with what is known about Egyptian mathematics from the time of construction of the pyramid, and with Egyptian theories of architecture and proportion used in their other works. | https://en.wikipedia.org/wiki?curid=12386 |
9,999 | The Parthenon's façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles. Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation." Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied." | https://en.wikipedia.org/wiki?curid=12386 |
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