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Ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for simpler activities, such as catch or juggling. Balls made from hard-wearing materials are used in engineering applications to provide very low friction bearings, known as ball bearings. Black-powder weapons use stone and metal balls as projectiles. Although many types of balls are today made from rubber, this form was unknown outside the Americas until after the voyages of Columbus. The Spanish were the first Europeans to see the bouncing rubber balls (although solid and not inflated) which were employed most notably in the Mesoamerican ballgame. Balls used in various sports in other parts of the world prior to Columbus were made from other materials such as animal bladders or skins, stuffed with various materials. As balls are one of the most familiar spherical objects to humans, the word "ball" may refer to or describe spherical or near-spherical objects.
"Ball" is used metaphorically sometimes to denote something spherical or spheroid, e.g., armadillos and human beings curl up into a ball, making a fist into a ball. Etymology. The first known use of the word "ball" in English in the sense of a globular body that is played with was in 1205 in "Layamon's Brut, or Chronicle of Britain" in the phrase, "" ("Some of them drove balls far across the fields.") The word came from the Middle English "bal" (inflected as "ball-e, -es"), in turn from Old Norse "böllr" (pronounced ; compare Old Swedish "baller", and Swedish "boll") from Proto-Germanic "ballu-z" (whence probably Middle High German "bal, ball-es", Middle Dutch "bal"), a cognate with Old High German "ballo, pallo", Middle High German balle from Proto-Germanic "ballon" (weak masculine), and Old High German "ballâ, pallâ", Middle High German "balle", Proto-Germanic "ballôn" (weak feminine). No Old English representative of any of these is known. (The answering forms in Old English would have been "beallu, -a, -e"—compare "bealluc, ballock".) If "ball-" was native in Germanic, it may have been a cognate with the Latin "foll-is" in sense of a "thing blown up or inflated." In the later Middle English spelling "balle" the word coincided graphically with the French "balle" "ball" and "bale" which has hence been erroneously assumed to be its source. French "balle" (but not "boule") is assumed to be of Germanic origin, itself, however. In Ancient Greek the word πάλλα ("palla") for "ball" is attested besides the word σφαίρα ("sfaíra"), "sphere".
History. Some form of game with a ball is found portrayed on Egyptian monuments. In Homer, Nausicaa was playing at ball with her maidens when Odysseus first saw her in the land of the Phaeacians (Od. vi. 100). And Halios and Laodamas performed before Alcinous and Odysseus with ball play, accompanied with dancing (Od. viii. 370). The most ancient balls in Eurasia have been discovered in Karasahr, China and are 3000 years old. They were made of hair-filled leather. Ancient Greeks. Among the ancient Greeks, games with balls (σφαῖραι) were regarded as a useful subsidiary to the more violent athletic exercises, as a means of keeping the body supple, and rendering it graceful, but were generally left to boys and girls. Of regular rules for the playing of ball games, little trace remains, if there were any such. The names in Greek for various forms, which have come down to us in such works as the Ὀνομαστικόν of Julius Pollux, imply little or nothing of such; thus, ἀπόρραξις ("aporraxis") only means the putting of the ball on the ground with the open hand, οὐρανία ("ourania"), the flinging of the ball in the air to be caught by two or more players; φαινίνδα ("phaininda") would seem to be a game of catch played by two or more, where feinting is used as a test of quickness and skill. Pollux (i. x. 104) mentions a game called episkyros (ἐπίσκυρος), which has often been looked on as the origin of football. It seems to have been played by two sides, arranged in lines; how far there was any form of "goal" seems uncertain. It was impossible to produce a ball that was perfectly spherical; children usually made their own balls by inflating pig's bladders and heating them in the ashes of a fire to make them rounder, although Plato (fl. 420s BC – 340s BC) described "balls which have leather coverings in twelve pieces".
Ancient Romans. Among the Romans, ball games were looked upon as an adjunct to the bath, and were graduated to the age and health of the bathers, and usually a place (sphaeristerium) was set apart for them in the baths (thermae). There appear to have been three types or sizes of ball, the pila, or small ball, used in catching games, the paganica, a heavy ball stuffed with feathers, and the follis, a leather ball filled with air, the largest of the three. This was struck from player to player, who wore a kind of gauntlet on the arm. There was a game known as trigon, played by three players standing in the form of a triangle, and played with the follis, and also one known as harpastum, which seems to imply a "scrimmage" among several players for the ball. These games are known to us through the Romans, though the names are Greek. Modern ball games. The various modern games played with a ball or balls and subject to rules are treated under their various names, such as polo, cricket, football, etc. Physics. In sports, many modern balls are pressurized. Some are pressurized at the factory (e.g. tennis, squash (sport)) and others are pressurized by users (e.g. volleyball, basketball, football). Almost all pressurized balls gradually leak air. If the ball is factory pressurized, there is usually a rule about whether the ball retains sufficient pressure to remain playable. Depressurized balls lack bounce and are often termed "dead". In extreme cases, a dead ball becomes flaccid. If the ball is pressured on use, there are generally rules about how the ball is pressurized before the match, and when (or whether) the ball can be repressurized or replaced.
Due to the ideal gas law, ball pressure is a function of temperature, generally tracking ambient conditions. Softer balls that are struck hard (especially squash balls) increase in temperature due to inelastic collision. In outdoor sports, wet balls play differently than dry balls. In indoor sports, balls may become damp due to hand sweat. Any form of humidity or dampness will affect a ball's surface friction, which will alter a player's ability to impart spin on the ball. The action required to apply spin to a ball is governed by the physics of angular momentum. Spinning balls travelling through air (technically a fluid) will experience the Magnus effect, which can produce lateral deflections in addition to the normal up-down curvature induced by a combination of wind resistance and gravity.
Binary relation In mathematics, a binary relation associates elements of one set called the "domain" with elements of another set called the "codomain". Precisely, a binary relation over sets formula_1 and formula_2 is a set of ordered pairs formula_3 where formula_4 is in formula_1 and formula_6 is in formula_2. It encodes the common concept of relation: an element formula_4 is "related" to an element formula_6, if and only if the pair formula_3 belongs to the set of ordered pairs that defines the binary relation. An example of a binary relation is the "divides" relation over the set of prime numbers formula_11 and the set of integers formula_12, in which each prime formula_13 is related to each integer formula_14 that is a multiple of formula_13, but not to an integer that is not a multiple of formula_13. In this relation, for instance, the prime number formula_17 is related to numbers such as formula_18, formula_19, formula_20, formula_21, but not to formula_22 or formula_23, just as the prime number formula_24 is related to formula_19, formula_20, and formula_23, but not to formula_28 or formula_29.
Binary relations, and especially homogeneous relations, are used in many branches of mathematics to model a wide variety of concepts. These include, among others: A function may be defined as a binary relation that meets additional constraints. Binary relations are also heavily used in computer science. A binary relation over sets formula_1 and formula_2 is an element of the power set of formula_32 Since the latter set is ordered by inclusion (formula_33), each relation has a place in the lattice of subsets of formula_32 A binary relation is called a "homogeneous relation" when formula_35. A binary relation is also called a "heterogeneous relation" when it is not necessary that formula_35. Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called "concepts", and placing them in a complete lattice.
In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox. A binary relation is the most studied special case formula_37 of an formula_38-ary relation over sets formula_39, which is a subset of the Cartesian product formula_40 Definition. Given sets formula_1 and formula_2, the Cartesian product formula_43 is defined as formula_44 and its elements are called "ordered pairs". A formula_45 over sets formula_1 and formula_2 is a subset of formula_32 The set formula_1 is called the or of formula_45, and the set formula_2 the or of formula_45. In order to specify the choices of the sets formula_1 and formula_2, some authors define a or as an ordered triple formula_55, where formula_56 is a subset of formula_43 called the of the binary relation. The statement formula_58 reads "formula_4 is formula_45-related to formula_6" and is denoted by formula_62. The or of formula_45 is the set of all formula_4 such that formula_62 for at least one formula_6. The "codomain of definition", , or of formula_45 is the set of all formula_6 such that formula_62 for at least one formula_4. The of formula_45 is the union of its domain of definition and its codomain of definition.
When formula_72 a binary relation is called a (or ). To emphasize the fact that formula_1 and formula_2 are allowed to be different, a binary relation is also called a heterogeneous relation. The prefix "hetero" is from the Greek ἕτερος ("heteros", "other, another, different"). A heterogeneous relation has been called a rectangular relation, suggesting that it does not have the square-like symmetry of a homogeneous relation on a set where formula_75 Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "... a variant of the theory has evolved that treats relations from the very beginning as or , i.e. as relations where the normal case is that they are relations between different sets." The terms "correspondence", dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product formula_43 without reference to formula_1 and formula_2, and reserve the term "correspondence" for a binary relation with reference to formula_1 and formula_2.
In a binary relation, the order of the elements is important; if formula_81 then formula_82 can be true or false independently of formula_62. For example, formula_24 divides formula_23, but formula_23 does not divide formula_24. Operations. Union. If formula_45 and formula_89 are binary relations over sets formula_1 and formula_2 then formula_92 is the of formula_45 and formula_89 over formula_1 and formula_2. The identity element is the empty relation. For example, formula_97 is the union of < and =, and formula_98 is the union of > and =. Intersection. If formula_45 and formula_89 are binary relations over sets formula_1 and formula_2 then formula_103 is the of formula_45 and formula_89 over formula_1 and formula_2. The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2". Composition. If formula_45 is a binary relation over sets formula_1 and formula_2, and formula_89 is a binary relation over sets formula_2 and formula_113 then formula_114 (also denoted by formula_115) is the of formula_45 and formula_89 over formula_1 and formula_113.
The identity element is the identity relation. The order of formula_45 and formula_89 in the notation formula_122 used here agrees with the standard notational order for composition of functions. For example, the composition (is parent of)formula_123(is mother of) yields (is maternal grandparent of), while the composition (is mother of)formula_123(is parent of) yields (is grandmother of). For the former case, if formula_4 is the parent of formula_6 and formula_6 is the mother of formula_14, then formula_4 is the maternal grandparent of formula_14. Converse. If formula_45 is a binary relation over sets formula_1 and formula_2 then formula_134 is the , also called , of formula_45 over formula_2 and formula_1. For example, formula_138 is the converse of itself, as is formula_139, and formula_140 and formula_141 are each other's converse, as are formula_97 and formula_143 A binary relation is equal to its converse if and only if it is symmetric. Complement. If formula_45 is a binary relation over sets formula_1 and formula_2 then formula_147 (also denoted by formula_148) is the of formula_45 over formula_1 and formula_2.
For example, formula_138 and formula_139 are each other's complement, as are formula_33 and formula_155, formula_156 and formula_157, formula_158 and formula_159, and for total orders also formula_140 and formula_98, and formula_141 and formula_97. The complement of the converse relation formula_164 is the converse of the complement: formula_165 If formula_72 the complement has the following properties: Restriction. If formula_45 is a binary homogeneous relation over a set formula_1 and formula_89 is a subset of formula_1 then formula_171 is the of formula_45 to formula_89 over formula_1. If formula_45 is a binary relation over sets formula_1 and formula_2 and if formula_89 is a subset of formula_1 then formula_180 is the of formula_45 to formula_89 over formula_1 and formula_2. If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions.
However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "formula_4 is parent of formula_6" to females yields the relation "formula_4 is mother of the woman formula_6"; its transitive closure does not relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother. Also, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers a property of the relation formula_97 is that every non-empty subset formula_190 with an upper bound in formula_191 has a least upper bound (also called supremum) in formula_192 However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation formula_97 to the rational numbers.
A binary relation formula_45 over sets formula_1 and formula_2 is said to be a relation formula_89 over formula_1 and formula_2, written formula_200 if formula_45 is a subset of formula_89, that is, for all formula_203 and formula_204 if formula_62, then formula_206. If formula_45 is contained in formula_89 and formula_89 is contained in formula_45, then formula_45 and formula_89 are called written formula_213. If formula_45 is contained in formula_89 but formula_89 is not contained in formula_45, then formula_45 is said to be than formula_89, written formula_220 For example, on the rational numbers, the relation formula_141 is smaller than formula_98, and equal to the composition formula_223. Matrix representation. Binary relations over sets formula_1 and formula_2 can be represented algebraically by logical matrices indexed by formula_1 and formula_2 with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over formula_1 and formula_2 and a relation over formula_2 and formula_113), the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. Homogeneous relations (when formula_35) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.
Types of binary relations. Some important types of binary relations formula_45 over sets formula_1 and formula_2 are listed below. Uniqueness properties: Totality properties (only definable if the domain formula_1 and codomain formula_2 are specified): Uniqueness and totality properties (only definable if the domain formula_1 and codomain formula_2 are specified): If relations over proper classes are allowed: Sets versus classes. Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory. For example, to model the general concept of "equality" as a binary relation formula_138, take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory. In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set formula_292, that contains all the objects of interest, and work with the restriction formula_293 instead of formula_138. Similarly, the "subset of" relation formula_33 needs to be restricted to have domain and codomain formula_296 (the power set of a specific set formula_292): the resulting set relation can be denoted by formula_298 Also, the "member of" relation needs to be restricted to have domain formula_292 and codomain formula_296 to obtain a binary relation formula_301 that is a set. Bertrand Russell has shown that assuming formula_158 to be defined over all sets leads to a contradiction in naive set theory, see "Russell's paradox".
Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory, and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple formula_55, as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.) With this definition one can for instance define a binary relation over every set and its power set. Homogeneous relation. A homogeneous relation over a set formula_1 is a binary relation over formula_1 and itself, i.e. it is a subset of the Cartesian product formula_306 It is also simply called a (binary) relation over formula_1. A homogeneous relation formula_45 over a set formula_1 may be identified with a directed simple graph permitting loops, where formula_1 is the vertex set and formula_45 is the edge set (there is an edge from a vertex formula_4 to a vertex formula_6 if and only if formula_62).
The set of all homogeneous relations formula_315 over a set formula_1 is the power set formula_317 which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on formula_315, it forms a semigroup with involution. Some important properties that a homogeneous relation formula_45 over a set formula_1 may have are: A is a relation that is reflexive, antisymmetric, and transitive. A is a relation that is irreflexive, asymmetric, and transitive. A is a relation that is reflexive, antisymmetric, transitive and connected. A is a relation that is irreflexive, asymmetric, transitive and connected. An is a relation that is reflexive, symmetric, and transitive. For example, "formula_4 divides formula_6" is a partial, but not a total order on natural numbers formula_358 "formula_359" is a strict total order on formula_358 and "formula_4 is parallel to formula_6" is an equivalence relation on the set of all lines in the Euclidean plane.
All operations defined in section also apply to homogeneous relations. Beyond that, a homogeneous relation over a set formula_1 may be subjected to closure operations like: Calculus of relations. Developments in algebraic logic have facilitated usage of binary relations. The calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations. The inclusion formula_200 meaning that formula_371 implies formula_372, sets the scene in a lattice of relations. But since formula_373 the inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of formula_374 In contrast to homogeneous relations, the composition of relations operation is only a partial function. The necessity of matching target to source of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations. The of the category Rel are sets, and the relation-morphisms compose as required in a category.
Induced concept lattice. Binary relations have been described through their induced concept lattices: A concept formula_375 satisfies two properties: For a given relation formula_380 the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion formula_381 forming a preorder. The MacNeille completion theorem (1937) (that any partial order may be embedded in a complete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices". The decomposition is Particular cases are considered below: formula_385 total order corresponds to Ferrers type, and formula_385 identity corresponds to difunctional, a generalization of equivalence relation on a set. Relations may be ranked by the Schein rank which counts the number of concepts necessary to cover a relation. Structural analysis of relations with concepts provides an approach for data mining. Particular relations. Difunctional. The idea of a difunctional relation is to partition objects by distinguishing attributes, as a generalization of the concept of an equivalence relation. One way this can be done is with an intervening set formula_399 of indicators. The partitioning relation formula_400 is a composition of relations using relations formula_401 Jacques Riguet named these relations difunctional since the composition formula_402 involves functional relations, commonly called "partial functions".
In 1950 Riguet showed that such relations satisfy the inclusion: In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This terminology recalls the fact that, when represented as a logical matrix, the columns and rows of a difunctional relation can be arranged as a block matrix with rectangular blocks of ones on the (asymmetric) main diagonal. More formally, a relation formula_45 on formula_43 is difunctional if and only if it can be written as the union of Cartesian products formula_406, where the formula_407 are a partition of a subset of formula_1 and the formula_409 likewise a partition of a subset of formula_2. Using the notation formula_411, a difunctional relation can also be characterized as a relation formula_45 such that wherever formula_413 and formula_414 have a non-empty intersection, then these two sets coincide; formally formula_415 implies formula_416 In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management." Furthermore, difunctional relations are fundamental in the study of bisimulations.
In the context of homogeneous relations, a partial equivalence relation is difunctional. Ferrers type. A strict order on a set is a homogeneous relation arising in order theory. In 1951 Jacques Riguet adopted the ordering of an integer partition, called a Ferrers diagram, to extend ordering to binary relations in general. The corresponding logical matrix of a general binary relation has rows which finish with a sequence of ones. Thus the dots of a Ferrer's diagram are changed to ones and aligned on the right in the matrix. An algebraic statement required for a Ferrers type relation R is formula_417 If any one of the relations formula_418 is of Ferrers type, then all of them are. Contact. Suppose formula_419 is the power set of formula_292, the set of all subsets of formula_292. Then a relation formula_384 is a contact relation if it satisfies three properties: The set membership relation, formula_426 "is an element of", satisfies these properties so formula_427 is a contact relation. The notion of a general contact relation was introduced by Georg Aumann in 1970.
In terms of the calculus of relations, sufficient conditions for a contact relation include formula_428 where formula_429 is the converse of set membership (formula_158). Preorder R\R. Every relation formula_45 generates a preorder formula_432 which is the left residual. In terms of converse and complements, formula_433 Forming the diagonal of formula_434, the corresponding row of formula_435 and column of formula_436 will be of opposite logical values, so the diagonal is all zeros. Then To show transitivity, one requires that formula_439 Recall that formula_440 is the largest relation such that formula_441 Then The inclusion relation Ω on the power set of formula_447 can be obtained in this way from the membership relation formula_158 on subsets of formula_447: Fringe of a relation. Given a relation formula_45, its fringe is the sub-relation defined as formula_452 When formula_45 is a partial identity relation, difunctional, or a block diagonal relation, then formula_454. Otherwise the formula_455 operator selects a boundary sub-relation described in terms of its logical matrix: formula_456 is the side diagonal if formula_45 is an upper right triangular linear order or strict order. formula_456 is the block fringe if formula_45 is irreflexive (formula_460) or upper right block triangular. formula_456 is a sequence of boundary rectangles when formula_45 is of Ferrers type.
On the other hand, formula_463 when formula_45 is a dense, linear, strict order. Mathematical heaps. Given two sets formula_292 and formula_419, the set of binary relations between them formula_467 can be equipped with a ternary operation formula_468 where formula_469 denotes the converse relation of formula_470. In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps, heaps, and generalized heaps. The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:
Braille Braille ( , ) is a tactile writing system used by blind or visually impaired people. It can be read either on embossed paper or by using refreshable braille displays that connect to computers and smartphone devices. Braille can be written using a slate and stylus, a braille writer, an electronic braille notetaker or with the use of a computer connected to a braille embosser. For blind readers, braille is an independent writing system, rather than a code of printed orthography. Braille is named after its creator, Louis Braille, a Frenchman who lost his sight as a result of a childhood accident. In 1824, at the age of fifteen, he developed the braille code based on the French alphabet as an improvement on night writing. He published his system, which subsequently included musical notation, in 1829. The second revision, published in 1837, was the first binary form of writing developed in the modern era. Braille characters are formed using a combination of six raised dots arranged in a 3 × 2 matrix, called the braille cell. The number and arrangement of these dots distinguishes one character from another. Since the various braille alphabets originated as transcription codes for printed writing, the mappings (sets of character designations) vary from language to language, and even within one; in English braille there are three levels: "uncontracted"a letter-by-letter transcription used for basic literacy; "contracted"an addition of abbreviations and contractions used as a space-saving mechanism; and "grade 3" various non-standardized personal stenographies that are less commonly used.
In addition to braille text (letters, punctuation, contractions), it is also possible to create embossed illustrations and graphs, with the lines either solid or made of series of dots, arrows, and bullets that are larger than braille dots. A full braille cell includes six raised dots arranged in two columns, each column having three dots. The dot positions are identified by numbers from one to six. There are 64 possible combinations, including no dots at all for a word space. Dot configurations can be used to represent a letter, digit, punctuation mark, or even a word. Early braille education is crucial to literacy, education and employment among the blind. Despite the evolution of new technologies, including screen reader software that reads information aloud, braille provides blind people with access to spelling, punctuation and other aspects of written language less accessible through audio alone. While some have suggested that audio-based technologies will decrease the need for braille, technological advancements such as braille displays have continued to make braille more accessible and available. Braille users highlight that braille remains as essential as print is to the sighted.
History. Braille was based on a tactile code, now known as night writing, developed by Charles Barbier. (The name "night writing" was later given to it when it was considered as a means for soldiers to communicate silently at night and without a light source, but Barbier's writings do not use this term and suggest that it was originally designed as a simpler form of writing and for the visually impaired.) In Barbier's system, sets of 12 embossed dots were used to encode 36 different sounds. Braille identified three major defects of the code: first, the symbols represented phonetic sounds and not letters of the alphabetthus the code was unable to render the orthography of the words. Second, the 12-dot symbols could not easily fit beneath the pad of the reading finger. This required the reading finger to move in order to perceive the whole symbol, which slowed the reading process. (This was because Barbier's system was based only on the number of dots in each of two 6-dot columns, not the pattern of the dots.) Third, the code did not include symbols for numerals or punctuation. Braille's solution was to use 6-dot cells and to assign a specific pattern to each letter of the alphabet. Braille also developed symbols for representing numerals and punctuation.
At first, braille was a one-to-one transliteration of the French alphabet, but soon various abbreviations (contractions) and even logograms were developed, creating a system much more like shorthand. Today, there are braille codes for over 133 languages. In English, some variations in the braille codes have traditionally existed among English-speaking countries. In 1991, work to standardize the braille codes used in the English-speaking world began. Unified English Braille (UEB) has been adopted in all seven member countries of the International Council on English Braille (ICEB) as well as Nigeria. Derivation. Braille is derived from the Latin alphabet, albeit indirectly. In Braille's original system, the dot patterns were assigned to letters according to their position within the alphabetic order of the French alphabet of the time, with accented letters and "w" sorted at the end. Unlike print, which consists of mostly arbitrary symbols, the braille alphabet follows a logical sequence. The first ten letters of the alphabet, "a"–"j", use the upper four dot positions: (black dots in the table below). These stand for the ten digits "1"–"9" and "0" in an alphabetic numeral system similar to Greek numerals (as well as derivations of it, including Hebrew numerals, Cyrillic numerals, Abjad numerals, also Hebrew gematria and Greek isopsephy).
Though the dots are assigned in no obvious order, the cells with the fewest dots are assigned to the first three letters (and lowest digits), "abc" = "123" (), and to the three vowels in this part of the alphabet, "aei" (), whereas the even digits "4", "6", "8", "0" () are right angles. The next ten letters, "k"–"t", are identical to "a"–"j" respectively, apart from the addition of a dot at position 3 (red dots in the bottom left corners of the cells in the table below): : The next ten letters (the next "decade") are the same again, but with dots also at both position 3 and position 6 (green dots in the bottom rows of the cells in the table above). Here "w" was left out as it was not part of the official French alphabet in Braille's time; the French order of the decade was "u v x y z ç é à è ù" (). The next ten letters, ending in "w", are the same again, except that for this series position 6 (purple dot in the bottom right corner of the cell in the table above) is used without a dot at position 3. In French braille these are the letters "â ê î ô û ë ï ü œ w" (). "W" had been tacked onto the end of 39 letters of the French alphabet to accommodate English.
The "a"–"j" series shifted down by one dot space () is used for punctuation. Letters "a" and "c" , which only use dots in the top row, were shifted two places for the apostrophe and hyphen: . (These are also the decade diacritics, on the left in the table below, of the second and third decade.) In addition, there are ten patterns that are based on the first two letters () with their dots shifted to the right; these were assigned to non-French letters ("ì ä ò" ), or serve non-letter functions: (superscript; in English the accent mark), (currency prefix), (capital, in English the decimal point), (number sign), (emphasis mark), (symbol prefix). The first four decades are similar in that the numeric sequence is extended by adding the decade dots, whereas in the fifth decade it is extended by shifting it downward. Originally there had been nine decades. The fifth through ninth used dashes as well as dots, but they proved to be impractical to distinguish by touch under normal conditions and were soon abandoned. From the beginning, these additional decades could be substituted with what we now know as the number sign () applied to the earlier decades, though that only caught on for the digits (the old 5th decade being replaced by applied to the 1st decade). The dash occupying the top row of the original sixth decade was simply omitted, producing the modern fifth decade. (See 1829 braille.)
Assignment. Historically, there have been three principles in assigning the values of a linear script (print) to Braille: Using Louis Braille's original French letter values; reassigning the braille letters according to the sort order of the print alphabet being transcribed; and reassigning the letters to improve the efficiency of writing in braille. Under international consensus, most braille alphabets follow the French sorting order for the 26 letters of the basic Latin alphabet, and there have been attempts at unifying the letters beyond these 26 (see international braille), though differences remain, for example, in German Braille. This unification avoids the chaos of each nation reordering the braille code to match the sorting order of its print alphabet, as happened in Algerian Braille, where braille codes were numerically reassigned to match the order of the Arabic alphabet and bear little relation to the values used in other countries (compare modern Arabic Braille, which uses the French sorting order), and as happened in an early American version of English Braille, where the letters "w", "x", "y", "z" were reassigned to match English alphabetical order. A convention sometimes seen for letters beyond the basic 26 is to exploit the physical symmetry of braille patterns iconically, for example, by assigning a reversed "n" to "ñ" or an inverted "s" to "sh". (See Hungarian Braille and Bharati Braille, which do this to some extent.)
A third principle was to assign braille codes according to frequency, with the simplest patterns (quickest ones to write with a stylus) assigned to the most frequent letters of the alphabet. Such frequency-based alphabets were used in Germany and the United States in the 19th century (see American Braille), but with the invention of the braille typewriter their advantage disappeared, and none are attested in modern use they had the disadvantage that the resulting small number of dots in a text interfered with following the alignment of the letters, and consequently made texts more difficult to read than Braille's more arbitrary letter assignment. Finally, there are braille scripts that do not order the codes numerically at all, such as Japanese Braille and Korean Braille, which are based on more abstract principles of syllable composition. Texts are sometimes written in a script of eight dots per cell rather than six, enabling them to encode a greater number of symbols. (See Gardner–Salinas braille codes.) Luxembourgish Braille has adopted eight-dot cells for general use; for example, accented letters take the unaccented versions plus dot 8.
Form. Braille was the first writing system with binary encoding. The system as devised by Braille consists of two parts: Within an individual cell, the dot positions are arranged in two columns of three positions. A raised dot can appear in any of the six positions, producing 64 (26) possible patterns, including one in which there are no raised dots. For reference purposes, a pattern is commonly described by listing the positions where dots are raised, the positions being universally numbered, from top to bottom, as 1 to 3 on the left and 4 to 6 on the right. For example, dot pattern 1-3-4 describes a cell with three dots raised, at the top and bottom in the left column and at the top of the right column: that is, the letter "m". The lines of horizontal braille text are separated by a space, much like visible printed text, so that the dots of one line can be differentiated from the braille text above and below. Different assignments of braille codes (or code pages) are used to map the character sets of different printed scripts to the six-bit cells. Braille assignments have also been created for mathematical and musical notation. However, because the six-dot braille cell allows only 64 (26) patterns, including space, the characters of a braille script commonly have multiple values, depending on their context. That is, character mapping between print and braille is not one-to-one. For example, the character corresponds in print to both the letter "d" and the digit "4".
In addition to simple encoding, many braille alphabets use contractions to reduce the size of braille texts and to increase reading speed. (See Contracted braille.) Writing braille. Braille may be produced by hand using a slate and stylus in which each dot is created from the back of the page, writing in mirror image, or it may be produced on a braille typewriter or Perkins Brailler, or an electronic Brailler or braille notetaker. Braille users with access to smartphones may also activate the on-screen braille input keyboard, to type braille symbols on to their device by placing their fingers on to the screen according to the dot configuration of the symbols they wish to form. These symbols are automatically translated into print on the screen. The different tools that exist for writing braille allow the braille user to select the method that is best for a given task. For example, the slate and stylus is a portable writing tool, much like the pen and paper for the sighted. Errors can be erased using a braille eraser or can be overwritten with all six dots (). "Interpoint" refers to braille printing that is offset, so that the paper can be embossed on both sides, with the dots on one side appearing between the divots that form the dots on the other.
Using a computer or other electronic device, Braille may be produced with a braille embosser (printer) or a refreshable braille display (screen). Eight-dot braille. Braille has been extended to an 8-dot code, particularly for use with braille embossers and refreshable braille displays. In 8-dot braille the additional dots are added at the bottom of the cell, giving a matrix 4 dots high by 2 dots wide. The additional dots are given the numbers 7 (for the lower-left dot) and 8 (for the lower-right dot). Eight-dot braille has the advantages that the casing of each letter is coded in the cell and that every printable ASCII character can be encoded in a single cell. All 256 (28) possible combinations of 8 dots are encoded by the Unicode standard. Braille with six dots is frequently stored as Braille ASCII. Letters. The first 25 braille letters, up through the first half of the 3rd decade, transcribe "a–z" (skipping "w"). In English Braille, the rest of that decade is rounded out with the ligatures "and, for, of, the," and "with". Omitting dot 3 from these forms the 4th decade, the ligatures "ch, gh, sh, th, wh, ed, er, ou, ow" and the letter "w".
Formatting. Various formatting marks affect the values of the letters that follow them. They have no direct equivalent in print. The most important in English Braille are: That is, is read as capital 'A', and as the digit '1'. Punctuation. Basic punctuation marks in English Braille include: is both the question mark and the opening quotation mark. Its reading depends on whether it occurs before a word or after. is used for both opening and closing parentheses. Its placement relative to spaces and other characters determines its interpretation. Punctuation varies from language to language. For example, French Braille uses for its question mark and swaps the quotation marks and parentheses (to and ); it uses () for both the period and the decimal point, and the English decimal point () to mark capitalization. Contractions. Braille contractions are words and affixes that are shortened so that they take up fewer cells. In English Braille, for example, the word "afternoon" is written with just three letters, , much like stenoscript. There are also several abbreviation marks that create what are effectively logograms. The most common of these is dot 5, which combines with the first letter of words. With the letter "m", the resulting word is "mother". There are also ligatures ("contracted" letters), which are single letters in braille but correspond to more than one letter in print. The letter "and", for example, is used to write words with the sequence "a-n-d" in them, such as "grand".
Page dimensions. Most braille embossers support between 34 and 40 cells per line, and 25 lines per page. A manually operated Perkins braille typewriter supports a maximum of 42 cells per line (its margins are adjustable), and typical paper allows 25 lines per page. A large interlining Stainsby has 36 cells per line and 18 lines per page. An A4-sized Marburg braille frame, which allows interpoint braille (dots on both sides of the page, offset so they do not interfere with each other), has 30 cells per line and 27 lines per page. Braille writing machine. A Braille writing machine is a typewriter with six keys that allows the user to write braille on a regular hard copy page. The first Braille typewriter to gain general acceptance was invented by Frank Haven Hall (Superintendent of the Illinois School for the Blind), and was presented to the public in 1892. The Stainsby Brailler, developed by Henry Stainsby in 1903, is a mechanical writer with a sliding carriage that moves over an aluminium plate as it embosses Braille characters. An improved version was introduced around 1933.
In 1951 David Abraham, a woodworking teacher at the Perkins School for the Blind, produced a more advanced Braille typewriter, the Perkins Brailler. Braille printers or embossers were produced in the 1950s. In 1960 Robert Mann, a teacher in MIT, wrote DOTSYS, a software that allowed automatic braille translation, and another group created an embossing device called "M.I.T. Braillemboss". The Mitre Corporation team of Robert Gildea, Jonathan Millen, Reid Gerhart and Joseph Sullivan (now president of Duxbury Systems) developed DOTSYS III, the first braille translator written in a portable programming language. DOTSYS III was developed for the Atlanta Public Schools as a public domain program. In 1991 Ernest Bate developed the Mountbatten Brailler, an electronic machine used to type braille on braille paper, giving it a number of additional features such as word processing, audio feedback and embossing. This version was improved in 2008 with a quiet writer that had an erase key. In 2011 David S. Morgan produced the first SMART Brailler machine, with added text to speech function and allowed digital capture of data entered.
Braille reading. Braille is traditionally read in hardcopy form, such as with paper books written in braille, documents produced in paper braille (such as restaurant menus), and braille labels or public signage. It can also be read on a refreshable braille display either as a stand-alone electronic device or connected to a computer or smartphone. Refreshable braille displays convert what is visually shown on a computer or smartphone screen into braille through a series of pins that rise and fall to form braille symbols. Currently more than 1% of all printed books have been translated into hardcopy braille. The fastest braille readers apply a light touch and read braille with two hands, although reading braille with one hand is also possible. Although the finger can read only one braille character at a time, the brain chunks braille at a higher level, processing words a digraph, root or suffix at a time. The processing largely takes place in the visual cortex. Literacy. Children who are blind miss out on fundamental parts of early and advanced education if not provided with the necessary tools, such as access to educational materials in braille. Children who are blind or visually impaired can begin learning foundational braille skills from a very young age to become fluent braille readers as they get older. Sighted children are naturally exposed to written language on signs, on TV and in the books they see. Blind children require the same early exposure to literacy, through access to braille rich environments and opportunities to explore the world around them. Print-braille books, for example, present text in both print and braille and can be read by sighted parents to blind children (and vice versa), allowing blind children to develop an early love for reading even before formal reading instruction begins.
Adults who experience sight loss later in life or who did not have the opportunity to learn it when they were younger can also learn braille. In most cases, adults who learn braille were already literate in print before vision loss and so instruction focuses more on developing the tactile and motor skills needed to read braille. While different countries publish statistics on how many readers in a given organization request braille, these numbers only provide a partial picture of braille literacy statistics. For example, this data does not survey the entire population of braille readers or always include readers who are no longer in the school system (adults) or readers who request electronic braille materials. Therefore, there are currently no reliable statistics on braille literacy rates, as described in a publication in the "Journal of Visual Impairment and Blindness". Regardless of the precise percentage of braille readers, there is consensus that braille should be provided to all those who benefit from it.
Numerous factors influence access to braille literacy, including school budget constraints, technology advancements such as screen-reader software, access to qualified instruction, and different philosophical views over how blind children should be educated. In the US, a key turning point for braille literacy was the passage of the Rehabilitation Act of 1973, an act of Congress that moved thousands of children from specialized schools for the blind into mainstream public schools. Because only a small percentage of public schools could afford to train and hire braille-qualified teachers, braille literacy has declined since the law took effect. Braille literacy rates have improved slightly since the bill was passed, in part because of pressure from consumers and advocacy groups that has led 27 states to pass legislation mandating that children who are legally blind be given the opportunity to learn braille. In 1998 there were 57,425 legally blind students registered in the United States, but only 10% (5,461) of them used braille as their primary reading medium.
Early Braille education is crucial to literacy for a blind or low-vision child. A study conducted in the state of Washington found that people who learned braille at an early age did just as well as, if not better than, their sighted peers in several areas, including vocabulary and comprehension. In the preliminary adult study, while evaluating the correlation between adult literacy skills and employment, it was found that 44% of the participants who had learned to read in braille were unemployed, compared to the 77% unemployment rate of those who had learned to read using print. Currently, among the estimated 85,000 blind adults in the United States, 90% of those who are braille-literate are employed. Among adults who do not know braille, only 33% are employed. Statistically, history has proven that braille reading proficiency provides an essential skill set that allows blind or low-vision children to compete with their sighted peers in a school environment and later in life as they enter the workforce. Regardless of the specific percentage of braille readers, proponents point out the importance of increasing access to braille for all those who can benefit from it.
Braille transcription. Although it is possible to transcribe print by simply substituting the equivalent braille character for its printed equivalent, in English such a character-by-character transcription (known as "uncontracted braille") is typically used by beginners or those who only engage in short reading tasks (such as reading household labels). Braille characters are much larger than their printed equivalents, and the standard page has room for only 25 lines of 43 characters. To reduce space and increase reading speed, most braille alphabets and orthographies use ligatures, abbreviations, and contractions. Virtually all English braille books in hardcopy (paper) format are transcribed in contracted braille: The Library of Congress's "Instruction Manual for Braille Transcribing" runs to over 300 pages, and braille transcribers must pass certification tests. Uncontracted braille was previously known as grade 1 braille, and contracted braille was previously known as grade 2 braille. Uncontracted braille is a direct transliteration of print words (one-to-one correspondence); hence, the word "about" would contain all the same letters in uncontracted braille as it does in inkprint. Contracted braille includes short forms to save space; hence, for example, the letters "ab" when standing alone represent the word "about" in English contracted braille. In English, some braille users only learn uncontracted braille, particularly if braille is being used for shorter reading tasks such as reading household labels. However, those who plan to use braille for educational and employment purposes and longer reading texts often go on to contracted braille.
The system of contractions in English Braille begins with a set of 23 words contracted to single characters. Thus the word "but" is contracted to the single letter "b", "can" to "c", "do" to "d", and so on. Even this simple rule creates issues requiring special cases; for example, "d" is, specifically, an abbreviation of the verb "do"; the noun "do" representing the note of the musical scale is a different word and must be spelled out. Portions of words may be contracted, and many rules govern this process. For example, the character with dots 2-3-5 (the letter "f" lowered in the Braille cell) stands for "ff" when used in the middle of a word. At the beginning of a word, this same character stands for the word "to"; the character is written in braille with no space following it. (This contraction was removed in the Unified English Braille Code.) At the end of a word, the same character represents an exclamation point. Some contractions are more similar than their print equivalents. For example, the contraction , meaning "letter", differs from , meaning "little", only by one dot in the second letter: "little", "letter". This causes greater confusion between the braille spellings of these words and can hinder the learning process of contracted braille.
The contraction rules take into account the linguistic structure of the word; thus, contractions are generally not to be used when their use would alter the usual braille form of a base word to which a prefix or suffix has been added. Some portions of the transcription rules are not fully codified and rely on the judgment of the transcriber. Thus, when the contraction rules permit the same word in more than one way, preference is given to "the contraction that more nearly approximates correct pronunciation". "Grade 3 braille" is a variety of non-standardized systems that include many additional shorthand-like contractions. They are not used for publication, but by individuals for their personal convenience. Braille translation software. When people produce braille, this is called braille transcription. When computer software produces braille, this is called a braille translator. Braille translation software exists to handle almost all of the common languages of the world, and many technical areas, such as mathematics (mathematical notation), for example WIMATS, music (musical notation), and tactile graphics.
Braille reading techniques. Since Braille is one of the few writing systems where tactile perception is used, as opposed to visual perception, a braille reader must develop new skills. One skill important for Braille readers is the ability to create smooth and even pressures when running one's fingers along the words. There are many different styles and techniques used for the understanding and development of braille, even though a study by B. F. Holland suggests that there is no specific technique that is superior to any other. Another study by Lowenfield & Abel shows that braille can be read "the fastest and best... by students who read using the index fingers of both hands". Another important reading skill emphasized in this study is to finish reading the end of a line with the right hand and to find the beginning of the next line with the left hand simultaneously. International uniformity. When Braille was first adapted to languages other than French, many schemes were adopted, including mapping the native alphabet to the alphabetical order of French – e.g. in English W, which was not in the French alphabet at the time, is mapped to braille X, X to Y, Y to Z, and Z to the first French-accented letter – or completely rearranging the alphabet such that common letters are represented by the simplest braille patterns. Consequently, mutual intelligibility was greatly hindered by this state of affairs. In 1878, the International Congress on Work for the Blind, held in Paris, proposed an international braille standard, where braille codes for different languages and scripts would be based, not on the order of a particular alphabet, but on phonetic correspondence and transliteration to Latin.
This unified braille has been applied to the languages of India and Africa, Arabic, Vietnamese, Hebrew, Russian, and Armenian, as well as nearly all Latin-script languages. In Greek, for example, γ (g) is written as Latin "g", despite the fact that it has the alphabetic position of "c"; Hebrew ב (b), the second letter of the alphabet and cognate with the Latin letter "b", is sometimes pronounced /b/ and sometimes /v/, and is written "b" or "v" accordingly; Russian ц (ts) is written as "c", which is the usual letter for /ts/ in those Slavic languages that use the Latin alphabet; and Arabic ف (f) is written as "f", despite being historically "p" and occurring in that part of the Arabic alphabet (between historic "o" and "q"). Other braille conventions. Other systems for assigning values to braille patterns are also followed beside the simple mapping of the alphabetical order onto the original French order. Some braille alphabets start with unified braille, and then diverge significantly based on the phonology of the target languages, while others diverge even further.
In the various Chinese systems, traditional braille values are used for initial consonants and the simple vowels. In both Mandarin and Cantonese Braille, however, characters have different readings depending on whether they are placed in syllable-initial (onset) or syllable-final (rime) position. For instance, the cell for Latin "k", , represents Cantonese "k" ("g" in Yale and other modern romanizations) when initial, but "aak" when final, while Latin "j", , represents Cantonese initial "j" but final "oei". Novel systems of braille mapping include Korean, which adopts separate syllable-initial and syllable-final forms for its consonants, explicitly grouping braille cells into syllabic groups in the same way as hangul. Japanese, meanwhile, combines independent vowel dot patterns and modifier consonant dot patterns into a single braille cell – an abugida representation of each Japanese mora. Uses. Braille is read by people who are blind, deafblind or who have low vision, and by both those born with a visual impairment and those who experience sight loss later in life. Braille may also be used by print impaired people, who although may be fully sighted, due to a physical disability are unable to read print. Even individuals with low vision will find that they benefit from braille, depending on level of vision or context (for example, when lighting or colour contrast is poor). Braille is used for both short and long reading tasks. Examples of short reading tasks include braille labels for identifying household items (or cards in a wallet), reading elevator buttons, accessing phone numbers, recipes, grocery lists and other personal notes. Examples of longer reading tasks include using braille to access educational materials, novels and magazines. People with access to a refreshable braille display can also use braille for reading email and ebooks, browsing the internet and accessing other electronic documents. It is also possible to adapt or purchase playing cards and board games in braille.
In India there are instances where the parliament acts have been published in braille, such as "The Right to Information Act". Sylheti Braille is used in Northeast India. In Canada, passenger safety information in braille and tactile seat row markers are required aboard planes, trains, large ferries, and interprovincial busses pursuant to the Canadian Transportation Agency's regulations. In the United States, the Americans with Disabilities Act of 1990 requires various building signage to be in braille. In the United Kingdom, medicines are required to have the name of the medicine in Braille on the labeling. Currency. The current series of Canadian banknotes has a tactile feature consisting of raised dots that indicate the denomination, allowing bills to be easily identified by blind or low vision people. It does not use standard braille numbers to identify the value. Instead, the number of full braille cells, which can be simply counted by both braille readers and non-braille readers alike, is an indicator of the value of the bill.
Mexican bank notes, Australian bank notes, Indian rupee notes, Israeli new shekel notes and Russian ruble notes also have special raised symbols to make them identifiable by persons who are blind or have low vision. Euro coins were designed in cooperation with organisations representing blind people, and as a result they incorporate many features allowing them to be distinguished by touch alone. In addition, their visual appearance is designed to make them easy to tell apart for persons who cannot read the inscriptions on the coins. "A good design for the blind and partially sighted is a good design for everybody" was the principle behind the cooperation of the European Central Bank and the European Blind Union during the design phase of the first series Euro banknotes in the 1990s. As a result, the design of the first euro banknotes included several characteristics which aid both the blind and partially sighted to confidently use the notes. Australia introduced the tactile feature onto their five-dollar banknote in 2016.
In the United Kingdom, the front of the £10 polymer note (the side with raised print), has two clusters of raised dots in the top left hand corner, and the £20 note has three. This tactile feature helps blind and partially sighted people identify the value of the note. In 2003 the US Mint introduced the commemorative Alabama State Quarter, which recognized State Daughter Helen Keller on the Obverse, including the name Helen Keller in both English script and Braille inscription. This appears to be the first known use of Braille on US Coin Currency, though not standard on all coins of this type. Unicode. The Braille set was added to the Unicode Standard in version 3.0 (1999). Most braille embossers and refreshable braille displays do not use the Unicode code points, but instead reuse the 8-bit code points that are assigned to standard ASCII for braille ASCII. (Thus, for simple material, the same bitstream may be interpreted equally as visual letter forms for sighted readers or their exact semantic equivalent in tactile patterns for blind readers. However some codes have quite different tactile versus visual interpretations and most are not even defined in Braille ASCII.)
Some embossers have proprietary control codes for 8-dot braille or for full graphics mode, where dots may be placed anywhere on the page without leaving any space between braille cells so that continuous lines can be drawn in diagrams, but these are rarely used and are not standard. The Unicode standard encodes 6-dot and 8-dot braille glyphs according to their binary appearance, rather than following their assigned numeric order. Dot 1 corresponds to the least significant bit of the low byte of the Unicode scalar value, and dot 8 to the high bit of that byte. The Unicode block for braille is U+2800 ... U+28FF. The mapping of patterns to characters etc. is language dependent: even for English for example, see American Braille and English Braille. Observation. Every year on 4 January, World Braille Day is observed internationally to commemorate the birth of Louis Braille and to recognize his efforts. Although the event is not considered a public holiday, it has been recognized by the United Nations as an official day of celebration since 2019. Braille devices. There is a variety of contemporary electronic devices that serve the needs of blind people that operate in Braille, such as refreshable braille displays and Braille e-books that use different technologies for transmitting graphic information of different types (pictures, maps, graphs, texts, etc.).
Bastille Day Bastille Day is the common name given in English-speaking countries to the national day of France, which is celebrated on 14 July each year. It is referred to, both legally and commonly, as () in French, though "la fête nationale" is also used in the press. French National Day is the anniversary of the Storming of the Bastille on 14 July 1789, a major event of the French Revolution, as well as the that celebrated the unity of the French people on 14 July 1790. Celebrations are held throughout France. One that has been reported as "the oldest and largest military parade in Europe" is held on 14 July on the Champs-Élysées in Paris in front of the President of France, along with other French officials and foreign guests. History. In 1789, tensions rose in France between reformist and conservative factions as the country struggled to resolve an economic crisis. In May, the Estates General legislative assembly was revived, but members of the Third Estate broke ranks, declaring themselves to be the National Assembly of the country, and on 20 June, vowed to write a constitution for the kingdom.
On 11 July, Jacques Necker, the finance minister of Louis XVI, who was sympathetic to the Third Estate, was dismissed by the King, provoking an angry reaction among Parisians. Crowds formed, fearful of an attack by the royal army or by foreign regiments of mercenaries in the King's service and seeking to arm themselves. Early on 14 July, a crowd besieged the Hôtel des Invalides for firearms, muskets, and cannons stored in its cellars. That same day, another crowd stormed the Bastille, a fortress-prison in Paris that had historically held people jailed on the basis of "lettres de cachet" (literally "signet letters"), arbitrary royal indictments that could not be appealed and did not indicate the reason for the imprisonment, and was believed to hold a cache of ammunition and gunpowder. As it happened, at the time of the attack, the Bastille held only seven inmates, none of great political significance. The crowd was eventually reinforced by the mutinous Régiment des Gardes Françaises ("Regiment of French Guards"), whose usual role was to protect public buildings. They proved a fair match for the fort's defenders, and Governor de Launay, the commander of the Bastille, capitulated and opened the gates to avoid a mutual massacre. According to the official documents, about 200 attackers and just one defender died before the capitulation. However, possibly because of a misunderstanding, fighting resumed. In this second round of fighting, de Launay and seven other defenders were killed, as was Jacques de Flesselles, the "prévôt des marchands" ("provost of the merchants"), the elected head of the city's guilds, who under the French monarchy had the responsibilities of a present-day mayor.
Shortly after the storming of the Bastille, late in the evening of 4 August, after a very stormy session of the "Assemblée constituante", feudalism was abolished. On 26 August, the Declaration of the Rights of Man and of the Citizen ("Déclaration des Droits de l'Homme et du Citoyen") was proclaimed. "Fête de la Fédération". As early as 1789, the year of the storming of the Bastille, preliminary designs for a national festival were underway. These designs were intended to strengthen the country's national identity through the celebration of the events of 14 July 1789. One of the first designs was proposed by Clément Gonchon, a French textile worker, who presented his design for a festival celebrating the anniversary of the storming of the Bastille to the French city administration and the public on 9 December 1789. There were other proposals and unofficial celebrations of 14 July 1789, but the official festival sponsored by the National Assembly was called the Fête de la Fédération. The "Fête de la Fédération" on 14 July 1790 was a celebration of the unity of the French nation during the French Revolution. The aim of this celebration, one year after the Storming of the Bastille, was to symbolize peace. The event took place on the Champ de Mars, which was located far outside of Paris at the time. The work needed to transform the Champ de Mars into a suitable location for the celebration was not on schedule to be completed in time. On the day recalled as the Journée des brouettes ("The Day of the Wheelbarrow"), thousands of Parisian citizens gathered together to finish the construction needed for the celebration.
The day of the festival, the National Guard assembled and proceeded along the boulevard du Temple in the pouring rain, and were met by an estimated 260,000 Parisian citizens at the Champ de Mars. A mass was celebrated by Talleyrand, bishop of Autun. The popular General Lafayette, as captain of the National Guard of Paris and a confidant of the king, took his oath to the constitution, followed by King Louis XVI. After the end of the official celebration, the day ended in a huge four-day popular feast, and people celebrated with fireworks, as well as fine wine and running nude through the streets in order to display their freedom. Origin of the current celebration. On 30 June 1878, a feast was officially arranged in Paris to honour the French Republic (the event was commemorated in a painting by Claude Monet). On 14 July 1879, there was another feast, with a semi-official aspect. The day's events included a reception in the Chamber of Deputies, organised and presided over by Léon Gambetta (a military reviewer at Longchamp), and a Republican Feast in the Pré Catelan. All throughout France, "Le Figaro" wrote, "people feasted much to honour the storming of the Bastille".
In 1880, the government of the Third Republic wanted to revive the 14 July festival. The campaign for the reinstatement of the festival was sponsored by the notable politician Léon Gambetta and scholar Henri Baudrillant. On 21 May 1880, Benjamin Raspail proposed a law, signed by sixty-four members of government, to have "the Republic adopt 14 July as the day of an annual national festival". There were many disputes over which date to be remembered as the national holiday, including 4 August (the commemoration of the end of the feudal system), 5 May (when the Estates-General first assembled), 27 July (the fall of Robespierre), and 21 January (the date of Louis XVI's execution). The government decided that the date of the holiday would be 14 July, but that was still somewhat problematic. The events of 14 July 1789 were illegal under the previous government, which contradicted the Third Republic's need to establish legal legitimacy. French politicians also did not want the sole foundation of their national holiday to be rooted in a day of bloodshed and class-hatred as the day of storming the Bastille was. Instead, they based the establishment of the holiday as both the celebration of the Fête de la Fédération, a festival celebrating the anniversary of the Republic of France on 14 July 1789, and the storming of the Bastille. The Assembly voted in favor of the proposal on 21 May, and 8 June. The law was approved on 27 and 29 June. The celebration was made official on 6 July 1880.
In the debate leading up to the adoption of the holiday, Senator Henri Martin, who wrote the National Day law, addressed the chamber on 29 June 1880: Bastille Day military parade. The Bastille Day military parade is the French military parade that has been held in the morning, every year in Paris, since 1880. While previously held elsewhere within or near the capital city, since 1918 it has been held on the Champs-Élysées, with the participation of the Allies as represented in the Versailles Peace Conference, and with the exception of the period of German occupation from 1940 to 1944 (when the ceremony took place in London under the command of General Charles de Gaulle); and 2020 when the COVID-19 pandemic forced its cancellation. The parade passes down the Champs-Élysées from the Arc de Triomphe to the Place de la Concorde, where the President of the French Republic, his government and foreign ambassadors to France stand. This is a popular event in France, broadcast on French TV, and is the oldest and largest regular military parade in Europe.
Smaller military parades are held in French garrison towns, including Toulon and Belfort, with local troops. Bastille Day celebrations in other countries. Belgium. Liège celebrates Bastille Day each year since the end of the First World War, as Liège was decorated by the Légion d'Honneur for its unexpected resistance during the Battle of Liège. The city also hosts a fireworks show outside of Congress Hall. Specifically in Liège, celebrations of Bastille Day have been known to be bigger than the celebrations of the Belgian National holiday. Around 35,000 people gather to celebrate Bastille Day. There is a traditional festival dance of the French consul that draws large crowds, and many unofficial events over the city celebrate the relationship between France and the city of Liège. Canada. Vancouver, British Columbia holds a celebration featuring exhibits, food and entertainment. The Toronto Bastille Day festival is also celebrated in Toronto, Ontario. The festival is organized by the French-Canadian community in Toronto and sponsored by the Consulate General of France. The celebration includes music, performances, sport competitions, and a French Market. At the end of the festival, there is also a traditional French bal populaire.
Czech Republic. Since 2008, Prague has hosted a French market "" ("Fourteenth of July Market") offering traditional French food and wine as well as music. The market takes place on Kampa Island, it is usually between 11 and 14 July. It acts as an event that marks the relinquish of the EU presidency from France to the Czech Republic. Traditional selections of French produce, including cheese, wine, meat, bread and pastries, are provided by the market. Throughout the event, live music is played in the evenings, with lanterns lighting up the square at night. Denmark. The amusement park Tivoli celebrates Bastille Day. Hungary. Budapest's two-day celebration is sponsored by the Institut de France. The festival is hosted along the Danube River, with streets filled with music and dancing. There are also local markets dedicated to French foods and wine, mixed with some traditional Hungarian specialties. At the end of the celebration, a fireworks show is held on the river banks. India. Bastille Day is celebrated with great festivity in Pondicherry, a former French colony.
Ireland. The Embassy of France in Ireland organizes several events around Dublin, Cork and Limerick for Bastille Day; including evenings of French music and tasting of French food. Many members of the French community in Ireland take part in the festivities. Events in Dublin include live entertainment, speciality menus on French cuisine, and screenings of popular French films. New Zealand. The Auckland suburb of Remuera hosts an annual French-themed Bastille Day street festival. Visitors enjoy mimes, dancers, music, as well as French foods and drinks. The budding relationship between the two countries, with the establishment of a Maori garden in France and exchange of their analyses of cave art, resulted in the creation of an official reception at the Residence of France. There is also an event in Wellington for the French community held at the Residence of France. South Africa. Franschhoek's weekend festival has been celebrated since 1993. (Franschhoek, or 'French Corner,' is situated in the Western Cape.) As South Africa's gourmet capital, French food, wine and other entertainment is provided throughout the festival. The French Consulate in South Africa also celebrates their national holiday with a party for the French community. Activities also include dressing up in different items of French clothing.
French Polynesia. Following colonial rule, France annexed a large portion of what is now French Polynesia. Under French rule, Tahitians were permitted to participate in sport, singing, and dancing competitions one day a year: Bastille Day. The single day of celebration evolved into the major Heiva i Tahiti festival in Papeete Tahiti, where traditional events such as canoe races, tattooing, and fire walks are held. The singing and dancing competitions continue with music composed with traditional instruments such as the nasal flute and ukulele. United Kingdom. Within the UK, London has a large French contingent, and celebrates Bastille Day at various locations across the city including Battersea Park, Camden Town and Kentish Town. Live entertainment is performed at Canary Wharf, with weeklong performances of French theatre at the Lion and Unicorn Theatre in Kentish Town. Restaurants feature cabarets and special menus across the city, and other celebrations include garden parties and sports tournaments. There is also a large event at the Bankside and Borough Market, where there is live music, street performers, and traditional French games played.
United States. The United States has over 20 cities that conduct annual celebrations of Bastille Day. The different cities celebrate with many French staples such as food, music, games, and sometimes the recreation of famous French landmarks. Baltimore, Maryland, has a large Bastille Day celebration each year at Petit Louis in the Roland Park area of Baltimore. Boston has a celebration annually, hosted by the French Cultural Center for 40 years. The street festival occurs in Boston's Back Bay neighborhood, near the Cultural Center's headquarters. The celebration includes francophone musical performers, dancing, and French cuisine. New York City has numerous Bastille Day celebrations each July, including "Bastille Day on 60th Street" hosted by the French Institute Alliance Française between Fifth and Lexington Avenues on the Upper East Side of Manhattan, Bastille Day on Smith Street in Brooklyn, and Bastille Day in Tribeca. There is also the annual Bastille Day Ball, taking place since 1924. Philadelphia's Bastille Day, held at Eastern State Penitentiary, involves Marie Antoinette throwing locally manufactured Tastykakes at the Parisian militia, as well as a re-enactment of the storming of the Bastille.
There is also the annual Bastille Day Ball, taking place since 1924. Philadelphia's Bastille Day, held at Eastern State Penitentiary, involves Marie Antoinette throwing locally manufactured Tastykakes at the Parisian militia, as well as a re-enactment of the storming of the Bastille. (This Philadelphia tradition ended in 2018.) In Newport, Rhode Island, the annual Bastille Day celebration is organized by the local chapter of the Alliance Française. It takes place at King Park in Newport at the monument memorializing the accomplishments of the General Comte de Rochambeau whose 6,000 to 7,000 French forces landed in Newport on 11 July 1780. Their assistance in the defeat of the English in the War of Independence is well documented and is proof of the special relationship between France and the United States. In Washington D.C., food, music, and auction events are sponsored by the Embassy of France. There is also a French Festival within the city, where families can meet period entertainment groups set during the time of the French Revolution. There is also a French Festival within the city, where families can meet period entertainment groups set during the time of the French Revolution. Restaurants host parties serving traditional French food.
In Dallas, Texas, the Bastille Day celebration, "Bastille On Bishop", began in 2010 and is held annually in the Bishop Arts District of the North Oak Cliff neighborhood, southwest of downtown just across the Trinity River. Dallas' French roots are tied to the short lived socialist Utopian community La Réunion, formed in 1855 and incorporated into the City of Dallas in 1860. Miami's celebration is organized by "French & Famous" in partnership with the French American Chamber of Commerce, the Union des Français de l'Etranger and many French brands. The event gathers over 1,000 attendees to celebrate "La Fête Nationale". The location and theme change every year. In 2017, the theme was "Guinguette Party" and attracted 1,200 francophiles at The River Yacht Club. New Orleans, Louisiana, has multiple celebrations, the largest in the historic French Quarter. In Austin, Texas, the Alliance Française d’Austin usually conducts a family-friendly Bastille Day party at the French Legation, the home of the French representative to the Republic of Texas from 1841 to 1845.
Chicago, Illinois, has hosted a variety of Bastille Day celebrations in a number of locations in the city, including Navy Pier and Oz Park. The recent incarnations have been sponsored in part by the Chicago branch of the French-American Chamber of Commerce and by the French Consulate-General in Chicago. Milwaukee's four-day street festival begins with a "Storming of the Bastille" with a 43-foot replica of the Eiffel Tower. Minneapolis, Minnesota, has a celebration with wine, French food, pastries, a flea market, circus performers and bands. Also in the Twin Cities area, the local chapter of the Alliance Française has hosted an annual event for years at varying locations with a competition for the "Best Baguette of the Twin Cities." Montgomery, Ohio, has a celebration with wine, beer, local restaurants' fare, pastries, games and bands. St. Louis, Missouri, has annual festivals in the Soulard neighborhood, the former French village of Carondelet, Missouri, and in the Benton Park neighborhood. The Chatillon-DeMenil Mansion in the Benton Park neighborhood, holds an annual Bastille Day festival with reenactments of the beheading of Marie Antoinette and Louis XVI, traditional dancing, and artillery demonstrations. Carondelet also began hosting an annual saloon crawl to celebrate Bastille Day in 2017. The Soulard neighborhood in St. Louis, Missouri celebrates its unique French heritage with special events including a parade, which honors the peasants who rejected the monarchy. The parade includes a 'gathering of the mob,' a walking and golf cart parade, and a mock beheading of the King and Queen.
Portland, Oregon, has celebrated Bastille Day with crowds up to 8,000, in public festivals at various public parks, since 2001. The event is coordinated by the Alliance Française of Portland. Seattle's Bastille Day celebration, held at the Seattle Center, involves performances, picnics, wine and shopping. Sacramento, California, conducts annual "waiter races" in the midtown restaurant and shopping district, with a street festival.
Blowfish (cipher) Blowfish is a symmetric-key block cipher, designed in 1993 by Bruce Schneier and included in many cipher suites and encryption products. Blowfish provides a good encryption rate in software, and no effective cryptanalysis of it has been found to date for smaller files. It is recommended Blowfish should not be used to encrypt files larger than 4GB in size, Twofish should be used instead. Blowfish has a 64-bit block size and therefore it could be vulnerable to Sweet32 birthday attacks. Schneier designed Blowfish as a general-purpose algorithm, intended as an alternative to the aging DES and free of the problems and constraints associated with other algorithms. At the time Blowfish was released, many other designs were proprietary, encumbered by patents, or were commercial or government secrets. Schneier has stated that "Blowfish is unpatented, and will remain so in all countries. The algorithm is hereby placed in the public domain, and can be freely used by anyone." Notable features of the design include key-dependent S-boxes and a highly complex key schedule.
The algorithm. Blowfish has a 64-bit block size and a variable key length from 32 bits up to 448 bits. It is a 16-round Feistel cipher and uses large key-dependent S-boxes. In structure it resembles CAST-128, which uses fixed S-boxes. The adjacent diagram shows Blowfish's encryption routine. Each line represents 32 bits. There are five subkey-arrays: one 18-entry P-array (denoted as K in the diagram, to avoid confusion with the Plaintext) and four 256-entry S-boxes (S0, S1, S2 and S3). Every round "r" consists of 4 actions: The F-function splits the 32-bit input into four 8-bit quarters and uses the quarters as input to the S-boxes. The S-boxes accept 8-bit input and produce 32-bit output. The outputs are added modulo 232 and XORed to produce the final 32-bit output (see image in the upper right corner). After the 16th round, undo the last swap, and XOR L with K18 and R with K17 (output whitening). Decryption is exactly the same as encryption, except that P1, P2, ..., P18 are used in the reverse order. This is not so obvious because xor is commutative and associative. A common misconception is to use inverse order of encryption as decryption algorithm (i.e. first XORing P17 and P18 to the ciphertext block, then using the P-entries in reverse order).
Blowfish's key schedule starts by initializing the P-array and S-boxes with values derived from the hexadecimal digits of pi, which contain no obvious pattern (see nothing up my sleeve number). The secret key is then, byte by byte, cycling the key if necessary, XORed with all the P-entries in order. A 64-bit all-zero block is then encrypted with the algorithm as it stands. The resultant ciphertext replaces P1 and P2. The same ciphertext is then encrypted again with the new subkeys, and the new ciphertext replaces P3 and P4. This continues, replacing the entire P-array and all the S-box entries. In all, the Blowfish encryption algorithm will run 521 times to generate all the subkeys about 4 KB of data is processed. Because the P-array is 576 bits long, and the key bytes are XORed through all these 576 bits during the initialization, many implementations support key sizes up to 576 bits. The reason for that is a discrepancy between the original Blowfish description, which uses 448-bit keys, and its reference implementation, which uses 576-bit keys. The test vectors for verifying third-party implementations were also produced with 576-bit keys. When asked which Blowfish version is the correct one, Bruce Schneier answered: "The test vectors should be used to determine the one true Blowfish".
Another opinion is that the 448 bits limit is present to ensure that every bit of every subkey depends on every bit of the key, as the last four values of the P-array don't affect every bit of the ciphertext. This point should be taken in consideration for implementations with a different number of rounds, as even though it increases security against an exhaustive attack, it weakens the security guaranteed by the algorithm. And given the slow initialization of the cipher with each change of key, it is granted a natural protection against brute-force attacks, which doesn't really justify key sizes longer than 448 bits. Blowfish in pseudocode. P[18] // "P-array of 18 elements" S[4][256] // "S-boxes: 4 arrays of 256 elements" function f(x): // "Calculates a function f on a 32-bit input x, using S-boxes and bit manipulation" high_byte := (x shifted right by 24 bits) second_byte := (x shifted right by 16 bits) AND 0xff third_byte := (x shifted right by 8 bits) AND 0xff low_byte := x AND 0xff h := S[0][high_byte] + S[1][second_byte]
return (h XOR S[2][third_byte]) + S[3][low_byte] procedure blowfish_encrypt(L, R): // "Encrypts two 32-bit halves L and R using the P-array and function f over 16 rounds" for round := 0 to 15: L := L XOR P[round] R := f(L) XOR R swap values of L and R swap values of L and R R := R XOR P[16] L := L XOR P[17] procedure blowfish_decrypt(L, R): // "Decrypts two 32-bit halves L and R using the P-array and function f over 16 rounds in reverse" for round := 17 down to 2: L := L XOR P[round] R := f(L) XOR R swap values of L and R swap values of L and R R := R XOR P[1] L := L XOR P[0] // "Initializes the P-array and S-boxes using the provided key, followed by key expansion" //" Initialize P-array with the key values" key_position := 0 for i := 0 to 17: k := 0 for j := 0 to 3: k := (k shifted left by 8 bits) OR key[key_position] key_position := (key_position + 1) mod key_length P[i] := P[i] XOR k //" Blowfish key expansion (521 iterations)" L := 0, R := 0 for i := 0 to 17 by 2: blowfish_encrypt(L, R)
P[i] := L P[i + 1] := R //" Fill S-boxes by encrypting L and R" for i := 0 to 3: for j := 0 to 255 by 2: blowfish_encrypt(L, R) S[i][j] := L S[i][j + 1] := R Blowfish in practice. Blowfish is a fast block cipher, except when changing keys. Each new key requires the pre-processing equivalent of encrypting about 4 kilobytes of text, which is very slow compared to other block ciphers. This prevents its use in certain applications, but is not a problem in others. Blowfish must be initialized with a key. It is good practice to have this key hashed with a hash function before use. In one application Blowfish's slow key changing is actually a benefit: the password-hashing method (crypt $2, i.e. bcrypt) used in OpenBSD uses an algorithm derived from Blowfish that makes use of the slow key schedule; the idea is that the extra computational effort required gives protection against dictionary attacks. "See" key stretching. Blowfish has a memory footprint of just over 4 kilobytes of RAM. This constraint is not a problem even for older desktop and laptop computers, though it does prevent use in the smallest embedded systems such as early smartcards.
Blowfish was one of the first secure block ciphers not subject to any patents and therefore freely available for anyone to use. This benefit has contributed to its popularity in cryptographic software. bcrypt is a password hashing function which, combined with a variable number of iterations (work "cost"), exploits the expensive key setup phase of Blowfish to increase the workload and duration of hash calculations, further reducing threats from brute force attacks. bcrypt is also the name of a cross-platform file encryption utility developed in 2002 that implements Blowfish. Weakness and successors. Blowfish's use of a 64-bit block size (as opposed to e.g. AES's 128-bit block size) makes it vulnerable to birthday attacks, particularly in contexts like HTTPS. In 2016, the SWEET32 attack demonstrated how to leverage birthday attacks to perform plaintext recovery (i.e. decrypting ciphertext) against ciphers with a 64-bit block size. The GnuPG project recommends that Blowfish not be used to encrypt files larger than 4 GB due to its small block size.
A reduced-round variant of Blowfish is known to be susceptible to known-plaintext attacks on reflectively weak keys. Blowfish implementations use 16 rounds of encryption, and are not susceptible to this attack. Bruce Schneier has recommended migrating to his Blowfish successor, Twofish. was released in 2005, developed by Alexander Pukall. It has exactly the same design but has twice as many S tables and uses 64-bit integers instead of 32-bit integers. It no longer works on 64-bit blocks but on 128-bit blocks like AES. Blowfish2 is used for example, in FreePascal.
Bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if and only if it is invertible; that is, a function formula_1 is bijective if and only if there is a function formula_2 the "inverse" of , such that each of the two ways for composing the two functions produces an identity function: formula_3 for each formula_4 in formula_5 and formula_6 for each formula_7 in formula_8 For example, the "multiplication by two" defines a bijection from the integers to the even numbers, which has the "division by two" as its inverse function. A function is bijective if and only if it is both injective (or "one-to-one")—meaning that each element in the codomain is mapped from at most one element of the domain—and surjective (or "onto")—meaning that each element of the codomain is mapped from at least one element of the domain. The term "one-to-one correspondence" must not be confused with "one-to-one function", which means injective but not necessarily surjective.
The elementary operation of counting establishes a bijection from some finite set to the first natural numbers , up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists a bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them. A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms its symmetric group. Some bijections with further properties have received specific names, which include automorphisms, isomorphisms, homeomorphisms, diffeomorphisms, permutation groups, and most geometric transformations. Galois correspondences are bijections between sets of mathematical objects of apparently very different nature. Definition. For a binary relation pairing elements of set "X" with elements of set "Y" to be a bijection, four properties must hold: Satisfying properties (1) and (2) means that a pairing is a function with domain "X". It is more common to see properties (1) and (2) written as a single statement: Every element of "X" is paired with exactly one element of "Y". Functions which satisfy property (3) are said to be "onto "Y" " and are called surjections (or "surjective functions"). Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or "injective functions"). With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto".
Examples. Batting line-up of a baseball or cricket team. Consider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set "X" will be the players on the team (of size nine in the case of baseball) and the set "Y" will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list. Seats and students of a classroom. In a classroom there are a certain number of seats. A group of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion was that:
The instructor was able to conclude that there were just as many seats as there were students, without having to count either set. Inverses. A bijection "f" with domain "X" (indicated by "f": "X → Y" in functional notation) also defines a converse relation starting in "Y" and going to "X" (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not, "in general", yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain "Y". Moreover, properties (1) and (2) then say that this inverse "function" is a surjection and an injection, that is, the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function is invertible if and only if it is a bijection. Stated in concise mathematical notation, a function "f": "X → Y" is bijective if and only if it satisfies the condition Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the batting order and outputs the player who will be batting in that position.
Composition. The composition formula_11 of two bijections "f": "X → Y" and "g": "Y → Z" is a bijection, whose inverse is given by formula_11 is formula_13. Conversely, if the composition formula_14 of two functions is bijective, it only follows that "f" is injective and "g" is surjective. Cardinality. If "X" and "Y" are finite sets, then there exists a bijection between the two sets "X" and "Y" if and only if "X" and "Y" have the same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of "same number of elements" (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets. Category theory. Bijections are precisely the isomorphisms in the category "Set" of sets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the category "Grp" of groups, the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are "group isomorphisms" which are bijective homomorphisms.
Generalization to partial functions. The notion of one-to-one correspondence generalizes to partial functions, where they are called "partial bijections", although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup. Another way of defining the same notion is to say that a partial bijection from "A" to "B" is any relation "R" (which turns out to be a partial function) with the property that "R" is the graph of a bijection "f":"A′"→"B′", where "A′" is a subset of "A" and "B′" is a subset of "B". When the partial bijection is on the same set, it is sometimes called a "one-to-one partial transformation". An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane. References. This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory. Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic may be found in any of these:
Binary function In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs. Precisely stated, a function formula_1 is binary if there exists sets formula_2 such that where formula_4 is the Cartesian product of formula_5 and formula_6 Alternative definitions. Set-theoretically, a binary function can be represented as a subset of the Cartesian product formula_7, where formula_8 belongs to the subset if and only if formula_9. Conversely, a subset formula_10 defines a binary function if and only if for any formula_11 and formula_12, there exists a unique formula_13 such that formula_8 belongs to formula_10. formula_16 is then defined to be this formula_17. Alternatively, a binary function may be interpreted as simply a function from formula_4 to formula_19. Even when thought of this way, however, one generally writes formula_16 instead of formula_21. Examples. Division of whole numbers can be thought of as a function. If formula_22 is the set of integers, formula_23 is the set of natural numbers (except for zero), and formula_24 is the set of rational numbers, then division is a binary function formula_25.
In a vector space "V" over a field "F", scalar multiplication is a binary function. A scalar "a" ∈ "F" is combined with a vector "v" ∈ "V" to produce a new vector "av" ∈ "V". Another example is that of inner products, or more generally functions of the form formula_26, where , are real-valued vectors of appropriate size and is a matrix. If is a positive definite matrix, this yields an inner product. Functions of two real variables. Functions whose domain is a subset of formula_27 are often also called functions of two variables even if their domain does not form a rectangle and thus the cartesian product of two sets. Restrictions to ordinary functions. In turn, one can also derive ordinary functions of one variable from a binary function. Given any element formula_11, there is a function formula_29, or formula_30, from formula_31 to formula_19, given by formula_33. Similarly, given any element formula_12, there is a function formula_35, or formula_36, from formula_5 to formula_19, given by formula_39. In computer science, this identification between a function from formula_4 to formula_19 and a function from formula_5 to formula_43, where formula_43 is the set of all functions from formula_31 to formula_19, is called "currying".
Generalisations. The various concepts relating to functions can also be generalised to binary functions. For example, the division example above is "surjective" (or "onto") because every rational number may be expressed as a quotient of an integer and a natural number. This example is "injective" in each input separately, because the functions "f" "x" and "f" "y" are always injective. However, it's not injective in both variables simultaneously, because (for example) "f" (2,4) = "f" (1,2). One can also consider "partial" binary functions, which may be defined only for certain values of the inputs. For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero. But this function is undefined when the second input is zero. A binary operation is a binary function where the sets "X", "Y", and "Z" are all equal; binary operations are often used to define algebraic structures. In linear algebra, a bilinear transformation is a binary function where the sets "X", "Y", and "Z" are all vector spaces and the derived functions "f" "x" and "f""y" are all linear transformations.
A bilinear transformation, like any binary function, can be interpreted as a function from "X" × "Y" to "Z", but this function in general won't be linear. However, the bilinear transformation can also be interpreted as a single linear transformation from the tensor product formula_47 to "Z". Generalisations to ternary and other functions. The concept of binary function generalises to "ternary" (or "3-ary") "function", "quaternary" (or "4-ary") "function", or more generally to "n-ary function" for any natural number "n". A "0-ary function" to "Z" is simply given by an element of "Z". One can also define an "A-ary function" where "A" is any set; there is one input for each element of "A". Category theory. In category theory, "n"-ary functions generalise to "n"-ary morphisms in a multicategory. The interpretation of an "n"-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original "n"-ary morphism will work in a monoidal category. The construction of the derived morphisms of one variable will work in a closed monoidal category. The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.
Blue Velvet (film) Blue Velvet is a 1986 American neo-noir mystery thriller film written and directed by David Lynch. Blending psychological horror with film noir, the film stars Kyle MacLachlan, Isabella Rossellini, Dennis Hopper, and Laura Dern, and is named after the 1951 song of the same name. The film follows a college student who returns to his hometown and discovers a severed human ear in a field, which leads him to uncover a criminal conspiracy involving a troubled nightclub singer. The screenplay of "Blue Velvet" had been passed around multiple times in the late 1970s and early 1980s, with several major studios declining it due to its strong sexual and violent content. After the failure of his 1984 film "Dune", Lynch made attempts at developing a more "personal story", somewhat characteristic of the surrealist style displayed in his first film "Eraserhead" (1977). The independent studio De Laurentiis Entertainment Group, owned at the time by Italian film producer Dino De Laurentiis, agreed to finance and produce the film.
"Blue Velvet" initially received a divided critical response, with many stating that its explicit content served little artistic purpose. Nevertheless, the film earned Lynch his second nomination for the Academy Award for Best Director, and received the year's Best Film and Best Director prizes from the National Society of Film Critics. It came to achieve cult status. As an example of a director casting against the norm, it was credited for revitalizing Hopper's career and for providing Rossellini with a dramatic outlet beyond her previous work as a fashion model and a cosmetics spokeswoman. In the years since, the film has been re-evaluated, and it is now widely regarded as one of Lynch's major works and one of the greatest films of the 1980s. Publications including "Sight & Sound", "Time", "Entertainment Weekly" and "BBC Magazine" have ranked it among the greatest American films of all time. In 2008, it was chosen by the American Film Institute as one of the ten greatest American mystery films. Plot. College student Jeffrey Beaumont returns to his suburban hometown of Lumberton, North Carolina, after his father, Tom, has a near-fatal attack from a medical condition. Walking home from the hospital, Jeffrey cuts through a vacant lot and discovers a severed human ear, which he takes to police detective John Williams. Williams's daughter Sandy tells Jeffrey that the ear somehow relates to a lounge singer named Dorothy Vallens. Intrigued, Jeffrey enters her apartment by posing as an exterminator. While there, he steals a spare key while she is distracted by a man in a distinctive yellow sport coat, whom Jeffrey nicknames the "Yellow Man".
Jeffrey and Sandy attend Dorothy's nightclub act, in which she sings "Blue Velvet", and leave early so Jeffrey can break into her apartment. Dorothy returns home and undresses; she finds Jeffrey hiding in a closet and forces him to strip at knifepoint, but he retreats to the closet when Frank Booth, a psychopathic gangster and drug lord, arrives and interrupts their encounter. Frank beats and rapes Dorothy while inhaling gas from a canister, alternating between fits of sobbing and violent rage. After Frank leaves, Jeffrey sneaks away and seeks comfort from Sandy. Surmising that Frank has abducted Dorothy's husband Don, and son Donnie, to force her into sexual slavery, Jeffrey suspects that Frank cut off Don's ear to intimidate her into submission. While continuing to see Sandy, Jeffrey enters into a sadomasochistic relationship with Dorothy, in which she encourages him to hit her. Jeffrey sees Frank attending Dorothy's show and later observes him selling drugs and meeting with the Yellow Man. Jeffrey then sees the Yellow Man meeting with a "well-dressed man".
When Frank catches Jeffrey leaving Dorothy's apartment, he abducts them and takes them to the lair of Ben, a criminal associate holding Don and Donnie hostage. Frank permits Dorothy to see her family and forces Jeffrey to watch Ben perform an impromptu lip-sync of Roy Orbison's "In Dreams", which moves Frank to tears. Afterwards, he and his gang take Jeffrey and Dorothy on a high-speed joyride to a sawmill yard, where he again attempts to sexually abuse Dorothy. When Jeffrey intervenes and punches him in the face, an enraged Frank and his gang pull him out of the car. Replaying the tape of "In Dreams", Frank smears lipstick on his face and violently kisses Jeffrey. Frank then has Jeffrey restrained and beats him unconscious, while Dorothy pleads for Frank to stop. Jeffrey awakens the next morning, bruised and bloodied. While visiting the police station, Jeffrey discovers that the Yellow Man is Detective Williams's partner Tom Gordon, who has been murdering Frank's rival drug dealers and stealing confiscated narcotics from the evidence room for Frank to sell. After Jeffrey and Sandy declare their love for each other at a party, they are pursued by a car which they assume belongs to Frank; as they arrive at Jeffrey's home, Sandy realizes the driver is her ex-boyfriend, Mike. After Mike threatens to beat Jeffrey for stealing his girlfriend, Dorothy appears on Jeffrey's porch naked, beaten, and confused. Mike backs down as Jeffrey and Sandy whisk Dorothy to Sandy's house to summon medical attention.
When Dorothy calls Jeffrey "my secret lover", a distraught Sandy slaps him for cheating on her. Jeffrey asks Sandy to tell her father everything, and Detective Williams then leads a police raid on Frank's headquarters, killing Frank's men. Jeffrey returns alone to Dorothy's apartment, where he discovers Don dead and Gordon mortally wounded. As Jeffrey leaves the apartment, the "Well-Dressed Man" arrives, sees Jeffrey in the stairs, and chases him back inside. Jeffrey uses Gordon's walkie-talkie to say he is in the bedroom before hiding in the closet. The "Well-Dressed Man" arrives at the apartment and Jeffrey observes he is actually Frank in disguise. Jeffrey kills Frank with Gordon's gun when Frank opens the closet door. Moments later, Sandy and Detective Williams arrive. Some time later, Jeffrey and Sandy have continued their relationship, Tom Beaumont has recovered, and Dorothy has been reunited with her son. Production. Origin. The film's story originated from three ideas that crystallized in the filmmaker's mind over a period of time starting as early as 1973.
The first idea was only "a feeling" and the title, as Lynch told "Cineaste" in 1987. The second idea was an image of a severed, human ear lying in a field. "I don't know why it had to be an ear. Except it needed to be an opening of a part of the body, a hole into something else ... The ear sits on the head and goes right into the mind so it felt perfect," Lynch remarked in a 1986 interview to "The New York Times". The third idea was Bobby Vinton's rendition of "Blue Velvet" and "the mood that came with that song a mood, a time, and things that were of that time." The scene in which Dorothy appears naked outside was inspired by a real-life experience Lynch had during childhood when he and his brother saw a naked woman walking down a neighborhood street at night. The experience was so traumatic to the young Lynch that it made him cry, and he had never forgotten it.
Casting. The cast of "Blue Velvet" included several then-relatively unknown actors. Lynch met Isabella Rossellini at a restaurant, and offered her the role of Dorothy Vallens. Helen Mirren had been Lynch's first choice for the role. Rossellini had gained some exposure before the film for her Lancôme ads in the early 1980s and for being the daughter of actress Ingrid Bergman and director Roberto Rossellini. After completion of the film, during test screenings, ICM Partners—the agency representing Rossellini—immediately dropped her as a client. Furthermore, the nuns at the school in Rome that Rossellini attended in her youth called to say they were praying for her. Kyle MacLachlan had played the central role in Lynch's critical and commercial failure "Dune" (1984), a science fiction epic based on the novel of the same name. MacLachlan later became a recurring collaborator with Lynch, who remarked: "Kyle plays innocents who are interested in the mysteries of life. He's the person you trust enough to go into a strange world with." Val Kilmer was offered a role in the film, but he turned it down as felt it was too "graphic" for him, a decision he later regretted. Dourif and Stockwell also rejoined Lynch from "Dune".
Dennis Hopper was the best-known actor in the film, having directed and starred in "Easy Rider" (1969). Hopper—said to be Lynch's third choice (Michael Ironside has stated that Frank was written with him in mind)—accepted the role, reportedly having exclaimed, "I've got to play Frank! I am Frank!" Harry Dean Stanton and Steven Berkoff both turned down the role of Frank because of the violent content in the film. Laura Dern, then 18 years old, was cast as Sandy after several already-successful actresses turned the role down, one among those being Molly Ringwald. Shooting. Principal photography of "Blue Velvet" began in August 1985 and completed in November. The film was shot at EUE/Screen Gems studio in Wilmington, North Carolina, which also provided the exterior scenes of Lumberton. The scene with a raped and battered Dorothy proved to be particularly challenging. Several townspeople arrived to watch the filming with picnic baskets and rugs, against the wishes of Rossellini and Lynch. However, they continued filming as normal, and when Lynch yelled cut, the townspeople left. As a result, police told Lynch they were no longer permitted to shoot in any public areas of Wilmington.
The Carolina Apartments in downtown Wilmington served as Dorothy's apartment building, with the adjacent Kenan fountain featured prominently in many shots. The building is also the birth place and death place of noted artist Claude Howell. The apartment building stands today, and the Kenan fountain was refurbished in 2020 after sustaining heavy damage during Hurricane Florence. Editing. Lynch's original rough cut ran for approximately four hours. He was contractually obligated to deliver a two-hour movie by De Laurentiis and cut many small subplots and character scenes. He also made cuts at the request of the MPAA. For example, when Frank slaps Dorothy after the first rape scene, the audience was supposed to see Frank actually hitting her. Instead, the film cuts away to Jeffrey in the closet, wincing at what he has just seen. This cut was made to satisfy the MPAA's concerns about violence, though Lynch thought that the change made the scene more disturbing. In 2011, Lynch announced that footage from the deleted scenes, long thought lost, had been discovered. The material was subsequently included on the Blu-ray Disc release of the film. Among the deleted footage was Megan Mullally as Jeffrey's college sweetheart Louise Wertham, whose entire role was cut from the theatrical release. The final cut of the film runs at just over two hours.
Distribution. Because the material was completely different from anything that would be considered mainstream at the time, De Laurentiis Entertainment Group's marketing employees were unsure of how to promote the film, or even if it would be promoted at all; it wasn't until the positive reception the film received at various film festivals that they began to promote it. Interpretations. Despite "Blue Velvet"s initial appearance as a mystery, the film operates on a number of thematic levels. The film owes a large debt to 1950s film noir, containing and exploring such conventions as the femme fatale (Dorothy Vallens), a seemingly unstoppable villain (Frank Booth) and the questionable moral outlook of the hero (Jeffrey Beaumont), as well as its unusual use of shadowy, sometimes dark cinematography. "Blue Velvet" establishes Lynch's famous "askew vision" and introduces several common elements of his work, some of which would later become his trademarks, including distorted characters, a polarized world and debilitating damage to the skull or brain. Perhaps the most significant Lynchian trademark in the film is the unearthing of a dark underbelly in a seemingly idealized small town; Jeffrey even proclaims in the film that he is "seeing something that was always hidden". Lynch's characterization of films, symbols and motifs have become well known and his particular style, characterised largely in "Blue Velvet" for the first time, has been written about extensively using descriptions like "dreamlike", "ultraweird", "dark", and "oddball". Red curtains also appear in key scenes, specifically in Dorothy's apartment and at the night club where she sings, which have since become a Lynch trademark. The film has been compared to Alfred Hitchcock's "Psycho" (1960) because of its stark treatment of evil and mental illness. The premise of both films is curiosity, leading to an investigation that draws the lead characters into a hidden, voyeuristic underworld of crime.
The film's thematic framework harks back to Edgar Allan Poe, Henry James and early gothic fiction, as well as films such as "Shadow of a Doubt" (1943) and "The Night of the Hunter" (1955) and the entire notion of film noir. Lynch has called it a "film about things that are hidden - within a small city and within people." Feminist psychoanalytic film theorist Laura Mulvey argues that "Blue Velvet" establishes a metaphorical Oedipal family—"the child", Jeffrey Beaumont and his "parents", Frank Booth and Dorothy Vallens - both through deliberate references to film noir. Michael Atkinson claims that the resulting violence in the film can be read as symbolic of domestic violence within real families. He reads Jeffrey as an innocent youth who is both horrified by the violence inflicted by Frank and tempted by it as the means of possessing Dorothy for himself. Atkinson takes a Freudian approach to the film, considering it to be an expression of the traumatised innocence which characterises Lynch's work. He states, "Dorothy represents the sexual force of the mother [figure] because she is forbidden and because she becomes the object of the unhealthy, infantile impulses at work in Jeffrey's subconscious."
Symbolism. Symbolism is used heavily in "Blue Velvet". The most consistent symbolism in the film is an insect motif introduced at the end of the first scene, when the camera zooms in on a well-kept suburban lawn until it unearths a swarming underground nest of bugs. This is generally recognized as a metaphor for the seedy underworld that Jeffrey will soon discover under the surface of his own suburban paradise. The severed ear he finds is being overrun by black ants. The bug motif is recurrent throughout the film, most notably in the bug-like gas mask that Frank wears and Jeffrey's exterminator disguise. One of Frank's accomplices is also consistently identified through the yellow jacket he wears, possibly referencing the name of a type of wasp. Finally, a robin eating a bug on a fence becomes a topic of discussion in the last scene of the film. The severed ear that Jeffrey discovers is also a key symbolic element, leading Jeffrey into danger. Indeed, just as Jeffrey's troubles begin, the audience is treated to a nightmarish sequence in which the camera zooms into the canal of the severed, decomposing ear. After the danger has subsided at the end the ear canal closeup is repeated, zooming out on a healthy and clean ear.
Soundtrack. The "Blue Velvet" soundtrack was supervised by Angelo Badalamenti (who makes a brief cameo appearance as the pianist at the Slow Club where Dorothy performs). The soundtrack makes heavy usage of vintage pop songs, such as Bobby Vinton's "Blue Velvet" and Roy Orbison's "In Dreams", juxtaposed with an orchestral score. During filming, Lynch placed speakers on set and in streets and played Shostakovich to set the mood he wanted to convey. The score alludes to Shostakovich's 15th Symphony, which Lynch had been listening to regularly while writing the screenplay. Lynch had originally opted to use "Song to the Siren" by This Mortal Coil during the scene in which Sandy and Jeffrey share a dance; however, he could not obtain the rights for the song at the time. He would go on to use this song in "Lost Highway" eleven years later. "Entertainment Weekly" ranked "Blue Velvet" soundtrack on its list of the "100 Greatest Film Soundtracks", at the 100th position. Critic John Alexander wrote, "the haunting soundtrack accompanies the title credits, then weaves through the narrative, accentuating the noir mood of the film." Lynch worked with music composer Angelo Badalamenti for the first time in this film and asked him to write a score that had to be "like Shostakovich, be very Russian, but make it the most beautiful thing but make it dark and a little bit scary." Badalamenti's success with "Blue Velvet" would lead him to contribute to all of Lynch's future full-length films until "Inland Empire" as well as the cult television program "Twin Peaks". Also included in the sound team was long-time Lynch collaborator Alan Splet, a sound editor and designer who had won an Academy Award for his work on "The Black Stallion" (1979) and been nominated for "Never Cry Wolf" (1983).
Reception. Box office. "Blue Velvet" premiered in competition at the Montreal World Film Festival in August 1986, and at the Toronto Festival of Festivals on September 12, 1986, and a few days later in the United States. It debuted commercially in both countries on September 19, 1986, in 98 theatres across the United States. In its opening weekend, the film grossed a total of $789,409. It eventually expanded to another 15 theatres, and in the US and Canada grossed a total of $8,551,228. "Blue Velvet" was met with uproar during its audience reception, with lines formed around city blocks in New York City and Los Angeles. There were reports of mass walkouts and refund demands during its opening week. At a Chicago screening, a man fainted and had to have his pacemaker checked. Upon completion, he returned to the cinema to see the ending. At a Los Angeles cinema, two strangers became engaged in a heated disagreement, but decided to resolve the disagreement to return to the theatre. Critical reception. "Blue Velvet" was released to a very polarized reception in the United States. The critics who did praise the film were often vociferous. "The New York Times" critic Janet Maslin directed much praise toward the performances of Hopper and Rossellini: "Mr. Hopper and Miss Rossellini are so far outside the bounds of ordinary acting here that their performances are best understood in terms of sheer lack of inhibition; both give themselves entirely over to the material, which seems to be exactly what's called for." She called it "an instant cult classic", concluding that "Blue Velvet" "is as fascinating as it is freakish" and "confirms Mr. Lynch's stature as an innovator, a superb technician, and someone best not encountered in a dark alley."
Sheila Benson of the "Los Angeles Times" called the film "the most brilliantly disturbing film ever to have its roots in small-town American life," describing it as "shocking, visionary, rapturously controlled". Film critic Gene Siskel included "Blue Velvet" on his list of the best films of 1986, at the fifth spot. Peter Travers, film critic for "Rolling Stone", named it the best film of the 1980s and referred to it as an "American masterpiece". Upon its initial release, Woody Allen and Martin Scorsese called "Blue Velvet" the best film of the year. On the other hand, Paul Attanasio of "The Washington Post" said "the film showcases a visual stylist utterly in command of his talents" and that Angelo Badalamenti "contributes an extraordinary score, slipping seamlessly from slinky jazz to violin figures to the romantic sweep of a classic Hollywood score," but stated that Lynch "isn't interested in communicating, he's interested in parading his personality. The movie doesn't progress or deepen, it just gets weirder, and to no good end." A general criticism from US critics was "Blue Velvet"s approach to sexuality and violence. They asserted that this detracted from the film's seriousness as a work of art, and some condemned the film as pornographic. One of its detractors, Roger Ebert, stated that the large amount of "jokey small-town satire" in the film made it impossible to take its themes seriously. Ebert praised Rossellini's performance as "convincing and courageous" but criticized how she was depicted in the film, even accusing David Lynch of misogyny: "degraded, slapped around, humiliated and undressed in front of the camera. And when you ask an actress to endure those experiences, you should keep your side of the bargain by putting her in an important film." While Ebert in later years came to consider Lynch a great filmmaker, his negative view of "Blue Velvet" remained unchanged after he revisited it in the 21st century.
The film is now widely considered a masterpiece and holds an approval score of 91% on Rotten Tomatoes based on 138 reviews, with an average rating of 8.2/10. The website's critical consensus states: "If audiences walk away from this subversive, surreal shocker not fully understanding the story, they might also walk away with a deeper perception of the potential of film storytelling." The film also has a score of 75 out of 100 on Metacritic based on 15 critics, indicating "generally favorable reviews". Looking back in his "Guardian/Observer" review, critic Philip French wrote, "The film is wearing well and has attained a classic status without becoming respectable or losing its sense of danger." Mark Kermode walked out on the film and gave the film a poor review upon its release, but revised his view of the film over time. In 2016, he remarked, "as a film critic, it taught me that when a film really gets under your skin and really provokes a visceral reaction, you have to be very careful about assessing it ... I didn't walk out on "Blue Velvet" because it was a bad film. I walked out on it because it was a really good film. The point was at the time I wasn't good enough for it."

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