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Order-5 120-cell Honeycomb
In the geometry of Hyperbolic space, hyperbolic 4-space, the order-5 120-cell honeycomb is one of five compact regular polytope, regular space-filling tessellations (or honeycomb (geometry), honeycombs). With Schläfli symbol , it has five 120-cells around each face. It is self-dual polytope, 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 *H.S.M. Coxeter, Coxeter, ''Regular Polytopes (book), Regular Polytopes'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes an ... [...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] |
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] |
Regular Polytopes (book)
''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a third edition by Dover Publications in 1973. The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries. Overview The main topics of the book are the Platonic solids (regular convex polyhedra), related polyhedra, and their higher-dimensional generalizations. It has 14 chapters, along with multiple appendices, providing a more complete treatment of the subject than any earlier work, and incorporating material from 18 of Coxeter's own previous papers. It includes many figures (both photographs of models by Paul Donchian and drawings), tables of numerical values, and historical remarks on the subject. The first chapter discusses regular polygons, regula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duoprism
In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, where and are dimensions of 2 (polygon) or higher. The lowest-dimensional duoprisms exist in 4-dimensional space as 4-polytopes being the Cartesian product of two polygons in 2-dimensional Euclidean space. More precisely, it is the Set (mathematics), set of points: :P_1 \times P_2 = \ where and are the sets of the points contained in the respective polygons. Such a duoprism is Convex polytope, convex if both bases are convex, and is bounded by prism (geometry), prismatic cells. Nomenclature Four-dimensional duoprisms are considered to be prismatic 4-polytopes. A duoprism constructed from two regular polygons of the same edge length is a uniform duoprism. A duoprism made of ''n''-polygons and ''m''-polygons is named by prefixing 'duopr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the ( convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , contai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex set, convex and non-convex shapes. Combinatorially equivalent to the regular octahedron The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it: * Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral. * Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 600-cell
In geometry, the Rectification (geometry), rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cell (mathematics), cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices. Containing the cell Hyperplane#Notes, realms of both the regular 120-cell and the regular 600-cell, it can be considered analogous to the polyhedron icosidodecahedron, which is a rectified icosahedron and rectified dodecahedron. The vertex figure of the rectified 600-cell is a uniform pentagonal prism. Semiregular polytope It is one of three Semiregular 4-polytopes#Semiregular polytopes, semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a ''octicosahedric'' for being made of octahedron and icosahedron cells. Emanuel Lodewijk Elte, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-5 Pentagonal Tiling
In geometry, the order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of , constructed from five pentagons around every vertex. As such, it is self-dual. Related tilings This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5n). See also *Square tiling *Uniform tilings in hyperbolic plane *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. ... References * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) * External links * * Hyperbolic and Spherical Tiling Gallery* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-5 Dodecahedral Honeycomb
In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular polytope, regular space-filling tessellations (or honeycomb (geometry), honeycombs) in Hyperbolic space, hyperbolic 3-space. With Schläfli symbol it has five regular dodecahedron, dodecahedral cells around each Edge (geometry), edge, and each Vertex (geometry), vertex is surrounded by twenty dodecahedra. Its vertex figure is an regular icosahedron, icosahedron. Description The dihedral angle of a Euclidean regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The regular dodecahedron, dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72� ... [...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. See also * List of regular polytopes References *Coxeter 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 ..., ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999 {{isbn, 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213) Honeycombs (geometry) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ..., ''The Beauty of Geometry: Twelve Essays'', Dover Pub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
