Symmetry
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together. Mathematical symmetry may be observed with respect to the passage of time; this article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people. The opposite of symmetry is asymmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion; this means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry if there is a line going through it which divides it into two pieces which are mirror images of each other.
An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has translational symmetry. An object has helical symmetry if it can be translated and rotated in three-dimensional space along a line known as a screw axis. An object contracted. Fractals exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection rotoreflection symmetry. A dyadic relation R is only if, whenever it's true that Rab, it's true that Rba. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary Mary is the same age as Paul. Symmetric binary logical connectives are and, or, nand and nor. Generalizing from geometrical symmetry in the previous section, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object; the set of operations that preserve a given property of the object form a group.
In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include and odd functions in calculus. In statistics, it appears as symmetric probability distributions, as skewness, asymmetry of distributions. Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations; this concept has become one of the most powerful tools of theoretical physics, as it has become evident that all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his read 1972 article More is Different that "it is only overstating the case to say that physics is the study of symmetry." See Noether's theorem. Important symmetries in physics include discrete symmetries of spacetime. In biology, the notion of symmetry is used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves.
Animals that move in one direction have upper and lower sides and tail ends, therefore a left and a right. The head becomes specialized with a mouth and sense organs, the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs remain asymmetric. Plants and sessile animals such as sea anemones have radial or rotational symmetry, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, sea lilies. In biology, the notion of symmetry is used as in physics, to say to describe the properties of the objects studied, including their interactions. A remarkable property of biological evolution is the changes of symmetry corresponding to the appearance of new parts and dynamics. Symmetry is important to chemistry because it undergirds all specific interactions between molecules in nature; the control of the symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer the
Quincunx
A quincunx is a geometric pattern consisting of five points arranged in a cross, with four of them forming a square or rectangle and a fifth at its center. It forms the arrangement of five units in the pattern corresponding to the five-spot on six-sided dice, playing cards, dominoes, it is represented in Unicode as U+2059 ⁙ FIVE DOT PUNCTUATION or U+2684 ⚄ DIE FACE-5. The quincunx was a coin issued by the Roman Republic c. 211–200 BC, whose value was five twelfths of an as, the Roman standard bronze coin. On the Roman quincunx coins, the value was sometimes indicated by a pattern of pellets. However, these dots were not always arranged in a quincunx pattern; the Oxford English Dictionary dates the first appearances of the Latin word in English as 1545 and 1574. The first citation for "A pattern used for planting trees" dates from 1606; the OED cites a 1647 reference to the German astronomer Kepler to the astronomical/astrological meaning. Jackson states that the word refers to the pattern of trees in an orchard, but uses it more abstractly for a version of the orchard-planting problem involving patterns of points and lines in the plane.
Quincunx patterns occur in many contexts: In heraldry, groups of five elements are arranged in a quincunx pattern, called in saltire in heraldic terminology. The flag of the Solomon Islands features this pattern, with its five stars representing the five main island groups in the Solomon Islands. Another instance of this pattern occurred in the flag of the 19th-century Republic of Yucatán, where it signified the five departments into which the republic was divided. A quincunx is a standard pattern for planting an orchard. Quincunxes are used in modern computer graphics as a pattern for multisample anti-aliasing. Quincunx antialiasing samples scenes at the corners and centers of each pixel; these five sample points, in the shape of a quincunx, are combined to produce each displayed pixel. However, samples at the corner points are shared with adjacent pixels, so the number of samples needed is only twice the number of displayed pixels. In numerical analysis, the quincunx pattern describes the two-dimensional five-point stencil, a sampling pattern used to derive finite difference approximations to derivatives.
In architecture, a quincuncial plan defined as a "cross-in-square", is the plan of an edifice composed of nine bays. The central and the four angular ones are covered with domes or groin vaults so that the pattern of these domes forms a quincunx. In Khmer architecture, the towers of a temple, such as Angkor Wat, are sometimes arranged in a quincunx to represent the five peaks of Mount Meru. A quincunx is one of the quintessential designs of Cosmatesque inlay stonework. A quincuncial map is a conformal map projection that maps the poles of the sphere to the centre and four corners of a square, thus forming a quincunx; the points on each face of a unit cell of a face-centred cubic lattice form a quincunx. The quincunx as a tattoo is known as the five dots tattoo, it has been variously interpreted as a fertility symbol, a reminder of sayings on how to treat women or police, a recognition symbol among the Romani people, a group of close friends, standing alone in the world, or time spent in prison.
Thomas Edison, whose many inventions included an electric pen which became the basis of a tattooing machine created by Samuel O'Reilly, had this pattern tattooed on his forearm. Republican Roman manipular legions adopted a checkered formation called quincunx when deployed for battle; the first two stages of the Saturn V super heavy-lift rocket had engines in a quincunx arrangement. A baseball diamond forms a quincunx with the pitcher's mound. Various literary works use or refer to the quincunx pattern for its symbolic value: The English physician Sir Thomas Browne in his philosophical discourse The Garden of Cyrus elaborates upon evidence of the quincunx pattern in art and mystically as evidence of "the wisdom of God". Although Browne wrote about quincunx in its geometric meaning, he may have been influenced by English astrology, as the astrological meaning of "quincunx" was introduced by the astronomer Kepler in 1604. James Joyce uses the term in Grace, a short story in The Dubliners of 1914, to describe the seating arrangement of five men in a church service.
Lobner argues that in this context the pattern serves as a symbol both of the wounds of Christ and of the Greek cross. Lawrence Durrell's novel sequence The Avignon Quintet is arranged in the form of a quincunx, according to the author; the Quincunx is the title of a lengthy and elaborate novel by Charles Palliser set in 19th-century England, published in 1989. In the first chapter of The Rings of Saturn, W. G. Sebald's narrator cites Browne's writing on the quincunx; the quincunx in turn becomes a model for the way. Séamus Heaney describes Ireland's historical provinces as together forming a quincunx, as the Irish word for province cúige explicates; the five provinces of Ireland were Ulster, Connacht and Meath. More in his essay Frontier
Marty Golubitsky
Martin Aaron Golubitsky is an American Distinguished professor of mathematics at Ohio State University and the former director of the Mathematical Biosciences Institute. Marty Golubitsky was born on April 1945 in Philadelphia, Pennsylvania, he graduated with bachelor's degree in 1966 from the University of Pennsylvania and the same year got his master's there as well. He obtained his Ph. D. from Massachusetts Institute of Technology in 1970 where his advisor was Victor Guillemin. From September 1974 to December 1976 he was an assistant professor at the Queens College and from January of next year to August 1979 served as an associate professor there. Starting from the same month of 1979 he relocated himself to the Arizona State University where he became a professor and served there till August 1983. In September of the same year he held the same position at the University of Houston where he remained till November 2008. From until 2016 he served as the director of the Mathematical Biosciences Institute at Ohio State University where he retains a distinguished professorship in mathematics.
He affiliates himself with such organizations as the American Association for the Advancement of Science, American Mathematical Society, Association for Women in Mathematics and Society for Industrial and Applied Mathematics. He served as the President of the Society for Industrial and Applied Mathematics 2005-2006. In 2012 he became an inaugural fellow of the American Mathematical Society, and in 2009 a SIAM Fellow. From January 1980 to June of the same year he worked at University of Nice Sophia Antipolis as a visiting professor and from September to December of the next year worked at the Duke University. Following that he worked at the University of California, with the same position which lasted him for two months in summer of 1982 and from January to June 1989 he worked at the Institute for Mathematics and Applications, a division of the University of Minnesota, he continued to hold that position four years when from January to June 1993 he was working at the division of University of Waterloo called Fields Institute.
From August to November 2005 he worked at both Newton Institute and Trinity College in Cambridge and from January to June 2006 worked at the University of Toronto as a distinguished professor. As of July 2005 he works as an adjunct professor at the Computational and Applied Mathematics division of Rice University. In 1992, he and Ian Stewart wrote a book called Fearful Symmetry: Is God a Geometer?, published by Blackwell Publishers in Oxford. In 1994 it was translated into Dutch by Hans van Cuijlenborg where it came out under a title of Turings tijger by Epsilon Uitgaven in Utrecht. In 1995 the same work was translated into Italian by Libero Sosio as Terribili simmetrie: Dio è un geometra? and was published in Turin, Italy. His second book, on which he worked with M. Field called Symmetry in Chaos: A Search for Pattern in Mathematics and Nature was released by Oxford University Press, in 1992 and was followed by German translation by Micha Lotrovsky in 1993 and the French one the same year by Christian Jeanmougin, published by Inter´Editions in Paris.
Besides books he has numerous peer-reviewed articles and was a co-editor of the Multiparameter Bifurcation Theory, Contemporary Mathematics, published by Association for Computing Machinery in 1986. Marty Golubitsky Marty Golubitsky
Ian Stewart (mathematician)
Ian Nicholas Stewart is a British mathematician and a popular-science and science-fiction writer. He is Emeritus Professor of Mathematics at the University of England. Stewart was born in 1945 in England. While in the sixth form at Harvey Grammar School in Folkestone he came to the attention of the mathematics teacher; the teacher had Stewart sit mock A-level examinations without any preparation along with the upper-sixth students. He was awarded a scholarship to study at the University of Cambridge as an undergraduate student of Churchill College, where he studied the Mathematical Tripos and obtained a first-class Bachelor of Arts degree in mathematics in 1966. Stewart went to the University of Warwick where his PhD on Lie algebras was supervised by Brian Hartley and completed in 1969. After his PhD, Stewart was offered an academic position at Warwick, he is well known for his popular expositions of mathematics and his contributions to catastrophe theory. While at Warwick, Stewart edited the mathematical magazine Manifold.
He wrote a column called "Mathematical Recreations" for Scientific American magazine from 1991 to 2001. This followed the work of past columnists like Martin Gardner, Douglas Hofstadter, A. K. Dewdney. Altogether, he wrote 96 columns for Scientific American, which were reprinted in the books "Math Hysteria", "How to Cut a Cake: And Other Mathematical Conundrums" and "Cows in the Maze". Stewart has held visiting academic positions in Germany, New Zealand, the US. Stewart has published more than 140 scientific papers, including a series of influential papers co-authored with Jim Collins on coupled oscillators and the symmetry of animal gaits. Stewart has collaborated with Jack Cohen and Terry Pratchett on four popular science books based on Pratchett's Discworld. In 1999 Terry Pratchett made both Jack Cohen and Professor Ian Stewart "Honorary Wizards of the Unseen University" at the same ceremony at which the University of Warwick gave Terry Pratchett an honorary degree. In March 2014 Ian Stewart's iPad app, Incredible Numbers by Professor Ian Stewart, launched in the App Store.
The app was produced in partnership with Touch Press. The Science of Discworld, with Jack Cohen and Terry Pratchett The Science of Discworld II: The Globe, with Jack Cohen and Terry Pratchett The Science of Discworld III: Darwin's Watch, with Jack Cohen and Terry Pratchett The Science of Discworld IV: Judgement Day, with Jack Cohen and Terry Pratchett Catastrophe Theory and its Applications, with Tim Poston, Pitman, 1978. ISBN 0-273-01029-8. Complex Analysis: The Hitchhiker's Guide to the Plane, I. Stewart, D Tall. 1983 ISBN 0-521-24513-3 Algebraic number theory and Fermat's last theorem, 3rd Edition, I. Stewart, D Tall. A. K. Peters ISBN 1-56881-119-5 Galois Theory, 3rd Edition and Hall ISBN 1-58488-393-6 Galois Theory Errata The Foundations of Mathematics, 2nd Edition, I. Stewart, D Tall. ISBN 978-019870-643-4 Wheelers, with Jack Cohen Heaven, with Jack Cohen, ISBN 0-446-52983-4, May 2004 In 1995 Stewart received the Michael Faraday Medal and in 1997 he gave the Royal Institution Christmas Lecture on The Magical Maze.
He was elected as a Fellow of the Royal Society in 2001. Stewart was the first recipient in 2008 of the Christopher Zeeman Medal, awarded jointly by the London Mathematical Society and the Institute of Mathematics and its Applications for his work on promoting mathematics. Stewart married his wife, Avril, in 1970, they met at a party at a house. They have two sons, he lists his recreations as science fiction, guitar, keeping fish, geology and snorkelling. Ian Stewart at the Mathematics Genealogy Project personal webpage Michael Faraday prize winners 2004–1986 Directory of Fellows of the Royal Society: Ian Stewart Prof Ian Stewart at Debrett's People of Today What does a Martian look like? Jack Cohen and Ian Stewart set out to find the answers Ian Stewart on space exploration by NASA Ian Stewart on Minesweeper one of the Millennium mathematics problems Press release about Terry Pratchett "Wizard Making" of Jack Cohen and Ian Stewart at the University of Warwick Interview with Ian Stewart on the Science of Discworld series Audio Interview with Ian Stewart on April 25, 2007 from WINA's Charlottesville Right Now Podcast series with Ian Stewart on the history of symmetry A Partly True Story published in: Scientific American, Feb 1993 "The Joy of Mathematics – A conversation with Ian Stewart", Ideas Roadshow, 2013 "In conversation with Ian Stewart", Chalkdust Magazine, 2016
Checkerboard
A checkerboard or chequerboard is a board of chequered pattern on which draughts is played. Most it consists of 64 squares of alternating dark and light color green and buff and red, or black and white. An 8×8 checkerboard is used to play many other games, including chess, whereby it is known as a chessboard. Other rectangular square-tiled boards are often called checkerboards. Given a matrix with m rows and n columns, a function f, f = { black if m ∧ 1 = n ∧ 1, white if m ∧ 1 ≠ n ∧ 1 or, alternatively, f = { black if m + n is white if m + n is odd The element = is black and represents the lower left corner of the board. Martin Gardner featured puzzles based on checkerboards in his November 1962 Mathematical Games column in Scientific American. A square checkerboard with an alternating pattern is used for games including: Amazons Chapayev Chess and some of its variants Czech draughts Draughts known as checkers Frisian draughts Gounki International draughts Italian draughts Lines of Action Pool checkers Russian checkersThe following games require an 8×8 board and are sometimes played on a chessboard.
Arimaa Breakthrough Crossings Mak-yek Makruk Martian Chess
Orbifold notation
In geometry, orbifold notation is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere, the frieze groups and wallpaper groups of the Euclidean plane, their analogues on the hyperbolic plane; the following types of Euclidean transformation can occur in a group described by orbifold notation: reflection through a line translation by a vector rotation of finite order around a point infinite rotation around a line in 3-space glide-reflection, i.e. reflection followed by translation. All translations which occur are assumed to form a discrete subgroup of the group symmetries being described.
Each group is denoted in orbifold notation by a finite string made up from the following symbols: positive integers 1, 2, 3, … the infinity symbol, ∞ the asterisk, * the symbol o, called a wonder and a handle because it topologically represents a torus closed surface. Patterns repeat by two translation; the symbol ×, called a miracle and represents a topological crosscap where a pattern repeats as a mirror image without crossing a mirror line. A string written in boldface represents a group of symmetries of Euclidean 3-space. A string not written in boldface represents a group of symmetries of the Euclidean plane, assumed to contain two independent translations; each symbol corresponds to a distinct transformation: an integer n to the left of an asterisk indicates a rotation of order n around a gyration point an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a kaleidoscopic point and reflects through a line an × indicates a glide reflection the symbol ∞ indicates infinite rotational symmetry around a line.
By abuse of language, we might say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way; the exceptional symbol o indicates that there are two linearly independent translations. An orbifold symbol is called good if it is not one of the following: p, pq, *p, *pq, for p,q>=2, p≠q. An object is chiral; the corresponding orbifold is non-orientable otherwise. The Euler characteristic of an orbifold can be read from its Conway symbol; each feature has a value: n without or before an asterisk counts as n − 1 n n after an asterisk counts as n − 1 2 n asterisk and × count as 1 o counts as 2. Subtracting the sum of these values from 2 gives the Euler characteristic. If the sum of the feature values is 2, the order is infinite, i.e. the notation represents a wallpaper group or a frieze group. Indeed, Conway's "Magic Theorem" indicates that the 17 wallpaper groups are those with the sum of the feature values equal to 2.
Otherwise, the order is 2 divided by the Euler characteristic. The following groups are isomorphic: 1* and *11 22 and 221 *22 and *221 2* and 2*1; this is. The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a third dimension to the object which does not add or spoil symmetry. For example, for a 2D image we can consider a piece of carton with that image displayed on one side, thus we have n• and *n•. The bullet is added on one- and two-dimensional groups to imply the existence of a fixed point. A 1D image can be drawn horizontally on a piece of carton, with a provision to avoid additional symmetry with respect to the line of the image, e.g. by drawing a horizontal bar under the image. Thus the discrete symmetry groups in one dimension are *•, *1•, ∞• and *∞•. Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the Cartesian product of the object and an asymmetric 2D or 1D object, respectively.
*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries §The diagram shows one fundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, 2-fold gyration points as small green squares. A first few hyperbolic groups, ordered by their Euler characteristic are: Mutation of orbifolds Fibrifold notation - an extension of orbifold notation for 3d space groups John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, W
John Horton Conway
John Horton Conway is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life. Conway spent the first half of his long career at the University of Cambridge, in England, the second half at Princeton University in New Jersey, where he now holds the title Professor Emeritus. Conway was born in the son of Cyril Horton Conway and Agnes Boyce, he became interested in mathematics at a early age. By the age of eleven his ambition was to become a mathematician. After leaving sixth form, Conway entered Caius College, Cambridge to study mathematics. Conway, a "terribly introverted adolescent" in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person: an "extrovert", he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport.
Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos, where he became an avid backgammon player, spending hours playing the game in the common room, he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University. Conway is known for the invention of the Game of Life, one of the early examples of a cellular automaton, his initial experiments in that field were done with pen and paper, long before personal computers existed. Since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, articles, it is a staple of recreational mathematics.
There is an extensive wiki devoted to cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data display. At times Conway has said he hates the Game of Life–largely because it has come to overshadow some of the other deeper and more important things he has done; the game did help launch a new branch of mathematics, the field of cellular automata. The Game of Life is now known to be Turing complete. Conway's career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner; when Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, over the years Gardner had written about recreational aspects of Conway's work. For instance, he discussed Conway's game of Sprouts and his angel and devil problem.
In the September 1976 column he reviewed Conway's book On Numbers and Games and introduced the public to Conway's surreal numbers. Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, Conway himself has been a featured speaker at these events, discussing various aspects of recreational mathematics. Conway is known for his contributions to combinatorial game theory, a theory of partisan games; this he developed with Elwyn Berlekamp and Richard Guy, with them co-authored the book Winning Ways for your Mathematical Plays. He wrote the book On Numbers and Games which lays out the mathematical foundations of CGT, he is one of the inventors of sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, Conway's soldiers, he came up with the angel problem, solved in 2006. He invented a new system of numbers, the surreal numbers, which are related to certain games and have been the subject of a mathematical novel by Donald Knuth.
He invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG. In the mid-1960s with Michael Guy, son of Richard Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms, they discovered the grand antiprism in the only non-Wythoffian uniform polychoron. Conway has suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation. In the theory of tessellations, he devised the Conway criterion which describes rules for deciding if a prototile will tile the plane, he investigated lattices in higher dimensions, was the first to determine the symmetry group of the Leech lattice. In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials.
Conway further developed tangle theory and invented a system of notation for tabulating knots, nowadays known as Conway notation, while correcting a number of errors in the 19th century knot tables and extending them to include all but four of the non-alternating primes with 11 crossings. See Topology Proceedings 7 118, he was the primary author of the ATLAS of Finite Groups giving prope