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120-cell Honeycomb
In the geometry of hyperbolic 4-space, the 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol , it has three 120-cells around each face. Its dual is the order-5 5-cell honeycomb, . Related honeycombs It is related to the order-4 120-cell honeycomb, , and order-5 120-cell honeycomb, . It is topologically similar to the finite 5-cube, , and 5-simplex, . It is analogous to the 120-cell, , and dodecahedron, . See also * List of regular polytopes References *Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) *Coxeter 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 ..., ''The Beauty of Geometry: Twelve Essays'', Dover Public ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Regular Polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an ''n''-polytope equivalently describes a tessellation of an (''n'' − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol , and with its octahedral symmetry, [4,3] or , it is represented by Coxeter diagram . The regular polytopes are ... [...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; the symmetry groups of regular polyhedra are an example. 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 Kac–Moody algebras. Standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-simplex
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The 5-simplex is a solution to the problem: ''Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.'' Alternate names It can also be called a hexateron, or hexa-5-tope, as a 6- facetted polytope in 5-dimensions. The name ''hexateron'' is derived from ''hexa-'' for having six facets and '' teron'' (with ''ter-'' being a corruption of ''tetra-'') for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the acronym hix. As a configuration This configuration matrix represents the 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-simplex. The nondiagonal numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, , around each cubic ridge. It can be called a penteract, a portmanteau of the Greek word , for 'five' (dimensions), and the word ''tesseract'' (the 4-cube). It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets. Related polytopes It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes. Applying an '' alternation'' operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes. The 5-cube can be seen as an ''order-3 tesseractic honeycomb'' on a 4-sphere. It is related to the Euclidea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-5 120-cell Honeycomb
In the geometry of hyperbolic 4-space, the order-5 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol , it has five 120-cells around each face. It is self- dual. It also has 600 120-cells around each vertex. Related honeycombs It is related to the (order-3) 120-cell honeycomb, and order-4 120-cell honeycomb. It is analogous to the order-5 dodecahedral honeycomb and order-5 pentagonal tiling. Birectified order-5 120-cell honeycomb The birectified order-5 120-cell honeycomb constructed by all rectified 600-cells, with octahedron and icosahedron cells, and triangle faces with a 5-5 duoprism vertex figure and has extended symmetry 5,3,3,5. See also * List of regular polytopes References *Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) *Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 &ndas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-4 120-cell Honeycomb
In the geometry of hyperbolic 4-space, the order-4 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol , it has four 120-cells around each face. Its dual is the order-5 tesseractic honeycomb, . Related honeycombs It is related to the (order-3) 120-cell honeycomb, and order-5 120-cell honeycomb. It is analogous to the order-4 dodecahedral honeycomb and order-4 pentagonal tiling. See also * List of regular polytopes References *Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) *Coxeter 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 ..., ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999 {{isbn, 0-486- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-5 5-cell Honeycomb
In the geometry of hyperbolic 4-space, the order-5 5-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol , it has five 5-cells around each face. Its dual is the 120-cell honeycomb, . Related honeycombs It is related to the order-5 tesseractic honeycomb, , and order-5 120-cell honeycomb, . It is topologically similar to the finite 5-orthoplex, , and 5-simplex, . It is analogous to the 600-cell, , and icosahedron, . See also * List of regular polytopes References *Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) *Coxeter 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 ..., ''The Beauty of Geometry: Twelve Essays'', Dover ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Polytope
In geometry, every polyhedron is associated with a second dual structure, where the Vertex (geometry), vertices of one correspond to the Face (geometry), faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or Abstract polytope, abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the Symmetry, symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is #Self-dual polyhedra, self-dual. The dual of an Isogonal figure, isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Honeycomb (geometry)
In geometry, a honeycomb is a ''space filling'' or ''close packing'' of polyhedral or higher-dimensional ''cells'', so that there are no gaps. It is an example of the more general mathematical ''tiling'' or ''tessellation'' in any number of dimensions. Its dimension can be clarified as ''n''-honeycomb for a honeycomb of ''n''-dimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. Classification There are infinitely many honeycombs, which have only been partially classified. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or ''slabs'' of prisms based on some tessellations of the plane. In particula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional spaces, 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. 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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension . Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians. Classically, a regular polytope in dimensions may be defined as having regular facets (-faces) and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes. A reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
