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3333 Symmetry
In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . Symmetry This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with six squares around every vertex. This symmetry by orbifold notation is called (*3333) with 4 order-3 mirror intersections. In Coxeter notation can be represented as ,4* removing two of three mirrors (passing through the square center) in the ,4symmetry. The *3333 symmetry can be doubled to 663 symmetry by adding a mirror bisecting the fundamental domain. This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction t1. A second 6-color symmetry can be constructed from a hexagonal symmetry domain. Example artwork Around 1956, M.C. Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a '' geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Disk Model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle. The group of orientation preserving isometries of the disk model is given by the projective special unitary group , the quotient of the special unitary group SU(1,1) by its center . Along with the Klein model and the Poincaré half-space model, it was proposed by Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry. It is named after Henri Poincaré, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami. The Poincaré ball model is the similar model for ''3'' or ''n''-dimensional hyperbolic geometry in which the points of the geometry ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isohedral Tilings
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same ''symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are eithe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Tilings
Isogonal is a mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ... term which means "having similar angles". It occurs in several contexts: * Isogonal polygon, polyhedron, polytope or tiling. * Isogonal trajectory in curve theory. * Isogonal conjugate in triangle geometry. An Isogonal is also the name for a line connecting points at which the magnetic declination is the same. {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Tilings
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they manifest hyperbolas, not because something about them is exaggerated. * Hyperbolic angle, an unbounded variable referring to a hyperbola instead of a circle * Hyperbolic coordinates, location by geometric mean and hyperbolic angle in quadrant I * Hyperbolic distribution, a probability distribution characterized by the logarithm of the probability density function being a hyperbola * Hyperbolic equilibrium point, a fixed point that does not have any center manifolds * Hyperbolic function, an analog of an ordinary trigonometric or circular function * Hyperbolic geometric graph, a random network generated by connecting nearby points sprinkled in a hyperbolic space * Hyperbolic geometry, a non-Euclidean geometry * Hyperbolic group, a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Tilings In Hyperbolic Plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry. Uniform tilings can be identified by their vertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents the heptagonal tiling which has 3 heptagons around each vertex. It is also regular since all the polygons are the same size, so it can also be given the Schläfli symbol . Uniform tilings may be regular (if also face- and edge-transitive), quasi-regular (if edge-transitive but not face-transitive) or semi-regular (if neither edge- nor face-transit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the hexagonal tiling. Uniform colorings There are 9 distinct uniform colorings of a square tiling. Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry. Three can be seen in the same symmetry domain as reduced colorings: 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii. Related polyhedra and tilings This tiling is topologically related as a part of sequence of regular polyhedra and tilings, extend ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts. A maternal cousin was the architect Sir Giles Gilbert Scott. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. Roberts, Siobhan, ''King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry'', Walker & Company, 2006, He felt that mathematics and music were intimately related, outlining his id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Escher Circlelimit Iv-with Overlay
Escher is a surname. Notable people with the surname include: * Alfred Escher (1819−1883), a Swiss politician and railway pioneer * Arnold Escher von der Linth (1807−1872), a Swiss geologist * Berend George Escher (1885−1967), a Dutch geologist, half-brother of M. C. Escher * George Arnold Escher (1843−1939), a Dutch civil engineer, foreign advisor to Japan, father of M. C. Escher * Gitta Escher (born 1957), a German gymnast * Hans Conrad Escher von der Linth (1767−1823), a Swiss scientist, civil engineer and politician * Heinrich Escher (1626−1710), a Swiss politician, mayor of Zürich * Josef Escher (1885−1954), a Swiss Federal Councilor * Luiz Jeferson Escher (born 1987), Brazilian footballer * Lydia Escher (1858−1891), a Swiss patron of the arts * M. C. Escher (Maurits Cornelis Escher; 1898−1972), a Dutch graphic artist * Rudolf George Escher (1912−1980), a Dutch composer and music theoretician * Sandra Escher (born 1945), a Dutch psychiatrist See al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyration Point
In geometry, a gyration is a rotation in a discrete subgroup of symmetries of the Euclidean plane such that the subgroup does not also contain a reflection symmetry whose axis passes through the center of rotational symmetry. In the orbifold corresponding to the subgroup, a gyration corresponds to a rotation point that does not lie on a mirror A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ..., called a gyration point. For example, having a sphere rotating about any point that is ''not'' the center of the sphere, the sphere is gyrating. If it was rotating about its center, the rotation would be symmetrical and it would not be considered gyration. References Euclidean geometry {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
