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Order-6 Cubic Honeycomb
In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron Cell (geometry), cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of Coxeter diagram#Paracompact (Koszul simplex groups), paracompact uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter diagrams for each family. These families can produce uniform honeycombs with infinite or unbounded Facet (geometry), facets or vertex figure, including ideal vertex, ideal vertices at infinity, similar to the Uniform tilings in hyperbolic plane#Ideal triangle domains, hyperbolic uniform tilings in two dimensions. Regular paracompact honeycombs Of the uniform paracompact H3 honeycombs, 11 are Honeycomb (geometry)#Uniform honeycombs, regular, meaning that their group of symmetries acts transitively on their flags. These have Schläfli symbol , , , , , , , , , , and , and are shown below. Four have finite Ideal polyhed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H3 336 CC Center
H3, H03 or H-3 may refer to: Entertainment * H3 (film), ''H3'' (film), a 2001 film about the 1981 Irish hunger strike * ''H3 – Halloween Horror Hostel'', a 2008 German horror-parody television film * ''Happy Hustle High'', a manga series by Rie Takada, originally titled "H3 School!" * h3h3Productions, styled "[h3]", a satirical YouTube channel Science * Triatomic hydrogen (H3), an unstable molecule * Trihydrogen cation (), one of the most abundant ions in the universe * Tritium (3H), or hydrogen-3, an isotope of hydrogen * ATC code H03 ''Thyroid therapy'', a subgroup of the Anatomical Therapeutic Chemical Classification System * British NVC community H3, a heath community of the British National Vegetation Classification system * Histamine H3 receptor, Histamine H3 receptor, a human gene * Histone H3, a component of DNA higher structure in eukaryotic cells * Hekla 3 eruption, a huge volcanic eruption around 1000 BC Computing * HTTP/3, the third major version of the Hype ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H3 344 CC Center
H3, H03 or H-3 may refer to: Entertainment * ''H3'' (film), a 2001 film about the 1981 Irish hunger strike * ''H3 – Halloween Horror Hostel'', a 2008 German horror-parody television film * ''Happy Hustle High'', a manga series by Rie Takada, originally titled "H3 School!" * h3h3Productions, styled " 3, a satirical YouTube channel Science * Triatomic hydrogen (H3), an unstable molecule * Trihydrogen cation (), one of the most abundant ions in the universe * Tritium (3H), or hydrogen-3, an isotope of hydrogen * ATC code H03 ''Thyroid therapy'', a subgroup of the Anatomical Therapeutic Chemical Classification System * British NVC community H3, a heath community of the British National Vegetation Classification system * Histamine H3 receptor, a human gene * Histone H3, a component of DNA higher structure in eukaryotic cells * Hekla 3 eruption, a huge volcanic eruption around 1000 BC Computing * HTTP/3, the third major version of the Hypertext Transfer Protocol * Socket H3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wythoff Construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process The method is based on the idea of tiling a sphere, with spherical triangles – see Schwarz triangles. This construction arranges three mirrors at the sides of a triangle, like in a kaleidoscope. However, different from a kaleidoscope, the mirrors are not parallel, but intersect at a single point. They therefore enclose a spherical triangle on the surface of any sphere centered on that point and repeated reflections produce a multitude of copies of the triangle. If the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. If one places a vertex at a suitable point inside the spherical triangle enclosed by the mirrors, it is possible to ensure that the reflections of that po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Diagram
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 books, including '' The Fifty-Nine Icosahedra'' (1938) and '' Regular Polytopes'' (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Biography Coxeter was born in Kensington, England, to Harold Samuel Coxeter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Space
In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of \mathbb R^n with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane. It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to distinguish it from complex hyperbolic spaces. Hyperbolic space serves as the prototype of a Gromov hyperbolic space, which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense. In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. (Here a "polygon" should be viewed as including the 2-dimensional region inside it.) Other names for a polygonal face include polyhedron side and Euclidean plane ''tile''. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Number of polygonal faces of a polyhedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polyhedron
In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform polyhedra may be Regular polyhedron, regular (if also Isohedral figure, face- and Isotoxal figure, edge-transitive), Quasiregular polyhedron, quasi-regular (if also edge-transitive but not face-transitive), or Semiregular polyhedron, semi-regular (if neither edge- nor face-transitive). The faces and vertices don't need to be Convex polyhedron, convex, so many of the uniform polyhedra are also Star polyhedron, star polyhedra. There are two infinite classes of uniform polyhedra, together with 75 other polyhedra. They are 2 infinite classes of Prism (geometry), prisms and antiprisms, the convex polyhedrons as in 5 Platonic solids and 13 Archimedean solids—2 Quasiregular polyhedron, quasiregular and 11 Semiregular polyhedron, semiregular&m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include '' regular tilings'' with regular polygonal tiles all of the same shape, and '' semiregular tilings'' with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An '' aperiodic tiling'' uses a small set of tile shapes that cannot form a repeating pattern (an aperiodic set of prototiles). A '' tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Honeycombs In Hyperbolic Space
In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedron, uniform polyhedral Cell (geometry), cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of Coxeter-Dynkin diagram#Application with uniform polytopes, rings of the Coxeter diagrams for each family. Hyperbolic uniform honeycomb families Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures (finite subgroups), and the second includes affine subgroups. Compact uniform honeycomb families The nine compact Coxeter groups are listed here with their Coxeter diagrams, in order of the relative volumes of their Fundamental domain, fundamental simplex domains.Felikson, 2002 These 9 families generate a total of 76 unique uniform honeycombs. The ... [...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] |
