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Alternated Order-4 Hexagonal Tiling
In geometry, the alternated order-4 hexagonal tiling is a Uniform tilings in hyperbolic plane, uniform tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of (3,4,4), h, and hr. Uniform constructions There are four uniform constructions, with some of lower ones which can be seen with two colors of triangles: Related polyhedra and tiling References * John Horton Conway, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) * See also *Square tiling *Uniform tilings in hyperbolic plane *List of regular polytopes External links * * Hyperbolic and Spherical Tiling Gallery * [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch] Hexagonal tilings Hyperbolic tilings Isogonal tilings Order-4 tilings Semiregular tilings {{hyperbolic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Tiling Verf 34343434
A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, security guards, in some workplaces and schools, and by inmates in prisons. In some countries, some other officials also wear uniforms in their duties; such is the case of the Commissioned Corps of the United States Public Health Service or the French prefects. For some organizations, such as police, it may be illegal for non-members to wear the uniform. Etymology From the Latin ''unus'' (meaning one), and ''forma'' (meaning form). Variants Corporate and work uniforms Workers sometimes wear uniforms or corporate clothing of one nature or another. Workers required to wear a uniform may include retail workers, bank and post-office workers, public-security and health-care workers, blue-collar employees, personal trainers in he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Tilings
Isogonal, a mathematical term meaning "having similar angles", may refer to: *Isogonal figure or polygon, polyhedron, polytope or tiling * Isogonal trajectory, in curve theory *Isogonal conjugate __NOTOC__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ..., in triangle geometry See also * Isogonic line, in the study of Earth's magnetic field, a line of constant magnetic declination {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Tilings
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 Face (geometry), faces and is vertex-transitive (Transitive group action, transitive on its vertex (geometry), vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are Congruence (geometry), congruent, and the tessellation, 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 polyhedron, regular (if also face- and edge-transitive), quasi- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal Tilings
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is defined as a hexagon that is both equilateral and equiangular. In other words, a hexagon is said to be regular if the edges are all equal in length, and each of its internal angle is equal to 120°. The Schläfli symbol denotes this polygon as \ . However, the regular hexagon can also be considered as the cutting off the vertices of an equilateral triangle, which can also be denoted as \mathrm\ . A regular hexagon is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac times the apothem (radius of the inscribed circle). Measurement The longest diagonals of a regu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Regular Polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. Overview This table shows a summary of regular polytope counts by rank. There are no Euclidean regular star tessellations in any number of dimensions. 1-polytopes There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol , or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a ''dion'' and gives it the Schläfli symbol . Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. 2-polytopes (polygons) The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A -gonal regular polygon is repre ... [...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 consisting of four squares around every vertex. John Horton Conway called it a quadrille. Structure and properties The square tiling has a structure consisting of one type of congruent prototile, the square, sharing two vertices with other identical ones. This is an example of monohedral tiling. Each vertex at the tiling is surrounded by four squares, which denotes in a vertex configuration as 4.4.4.4 or 4^4 . The vertices of a square can be considered as the lattice, so the square tiling can be formed through the square lattice. This tiling is commonly familiar with the flooring and game boards. It is self-dual, meaning the center of each square connects to another of the adjacent tile, forming square tiling itself. The square tiling acts transitively on the ''flags'' of the tiling. In this case, the flag consists of a mutually incident vertex, edge, and tile ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was 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 Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2 Tiling 344-1
H, or h, is the eighth letter of the Latin alphabet, used in the modern English alphabet, including the alphabets of other western European languages and others worldwide. Its name in English is ''aitch'' (pronounced , plural ''aitches''), or regionally ''haitch'' (pronounced , plural ''haitches'')''.''"H" ''Oxford English Dictionary,'' 2nd edition (1989); ''Merriam-Webster's Third New International Dictionary of the English Language, Unabridged'' (1993); "aitch" or "haitch", op. cit. Name English For most English speakers, the name for the letter is pronounced as and spelled "aitch" or occasionally "eitch". The pronunciation and the associated spelling "haitch" are often considered to be h-adding and are considered non-standard in England. It is, however, a feature of Hiberno-English, and occurs sporadically in various other dialects. The perceived name of the letter affects the choice of indefinite article before initialisms beginning with H: for example "an H-bomb" ... [...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 Face (geometry), faces and is vertex-transitive (Transitive group action, transitive on its vertex (geometry), vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are Congruence (geometry), congruent, and the tessellation, 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 polyhedron, regular (if also face- and edge-transitive), quasi- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3222 Symmetry
In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr. It can be seen as constructed as a rectified tetrahexagonal tiling, r, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling. Constructions There are two uniform constructions of this tiling, one from ,4or (*642) symmetry, and secondly removing the mirror middle, ,1+,4 gives a rectangular fundamental domain ˆž,3,∞ (*3222). There are 3 lower symmetry forms seen by including edge-colorings: sees the hexagons as truncated triangles, with two color edges, with ,4+(4*3) symmetry. sees the yellow squares as rectangles, with two color edges, with +,4(6*2) symmetry. A final quarter symmetry combines these colorings, with +,4+(32×) symmetry, with 2 and 3 fold gyration points and glide reflections. This four color tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
