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Quarter 6-cubic Honeycomb
In Six-dimensional space, six-dimensional Euclidean geometry, the quarter 6-cubic honeycomb is a uniform space-filling tessellation (or honeycomb (geometry), honeycomb). It has half the vertices of the 6-demicubic honeycomb, and a quarter of the vertices of a 6-cube honeycomb.Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318 Its facets are 6-demicubes, stericated 6-demicubes, and × duoprisms. Related honeycombs See also Regular and uniform honeycombs in 5-space: *6-cube honeycomb *6-demicube honeycomb * 6-simplex honeycomb * Truncated 6-simplex honeycomb * Omnitruncated 6-simplex honeycomb Notes References * Kaleidoscopes: Selected Writings of Harold Scott MacDonald Coxeter, H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] See p31 * {{Honeycombs Honeycombs (geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 6-honeycomb
In seven-dimensional space, seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope Ridge (geometry), ridge being shared by exactly two 6-polytope Facet (mathematics), facets. A uniform 7-polytope is one whose symmetry group is vertex-transitive, transitive on vertices and whose facets are uniform 6-polytopes. Regular 7-polytopes Regular 7-polytopes are represented by the Schläfli symbol with u 6-polytopes Facet (mathematics), facets around each 4-face. There are exactly three such List of regular polytopes#Convex 4, convex regular 7-polytopes: # - 7-simplex # - 7-cube # - 7-orthoplex There are no nonconvex regular 7-polytopes. Characteristics The topology of any given 7-polytope is defined by its Betti numbers and torsion coefficient (topology), torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the Euler characteristic used to characteri ... [...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] |
Harold Scott MacDonald 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 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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omnitruncated 6-simplex Honeycomb
In Six-dimensional space, six-dimensional Euclidean geometry, the omnitruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb (geometry), honeycomb). It is composed entirely of omnitruncated 6-simplex facets. The facets of all omnitruncated simplectic honeycombs are called permutahedron, permutahedra and can be positioned in ''n+1'' space with integral coordinates, permutations of the whole numbers (0,1,..,n). A lattice The A lattice (also called A) is the union of seven A6 lattice, A6 lattices, and has the vertex arrangement of the dual to the ''omnitruncated 6-simplex honeycomb'', and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex. : ∪ ∪ ∪ ∪ ∪ ∪ = dual of Related polytopes and honeycombs See also Regular and uniform honeycombs in 6-space: *6-cubic honeycomb *6-demicubic honeycomb *6-simplex honeycomb *Truncated 6-simplex honeycomb *2_22 honeycomb, 222 honeycomb Notes References * Norman Johnso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 6-simplex Honeycomb
In six-dimensional Euclidean geometry, the cyclotruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb. Structure It can be constructed by seven sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-simplex honeycomb divisions on each hyperplane. Related polytopes and honeycombs See also Regular and uniform honeycombs in 6-space: *6-cubic honeycomb *6-demicubic honeycomb * 6-simplex honeycomb *Omnitruncated 6-simplex honeycomb In Six-dimensional space, six-dimensional Euclidean geometry, the omnitruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb (geometry), honeycomb). It is composed entirely of omnitruncated 6-simplex facets. The facets of all ... * 222 honeycomb Notes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-simplex Honeycomb
In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb. A6 lattice This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the _6 Coxeter group. It is the 6-dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6-simplex, 21+21 rectified 6-simplex, 35+35 birectified 6-simplex, with the count distribution from the 8th row of Pascal's triangle. The A lattice (also called A) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex. : ∪ ∪ ∪ ∪ ∪ ∪ = dual of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-demicube Honeycomb
The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb. It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h and the alternated vertices create 6-orthoplex facets. D6 lattice The vertex arrangement of the 6-demicubic honeycomb is the D6 lattice. The 60 vertices of the rectified 6-orthoplex vertex figure of the ''6-demicubic honeycomb'' reflect the kissing number 60 of this lattice. The best known is 72, from the E6 lattice and the 222 honeycomb. The D lattice (also called D) can be constructed by the union of two D6 lattices. This packing is only a lattice for even dimensions. The kissing number is 25=32 (2n-1 for n8). : ∪ The D lattice (also called D and C) can be constructed by the union of all four 6-demicubic lattices: It is also the 6-dimensional body centered cubic, the union of ... [...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 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 if both bases are convex, and is bounded by 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 'duoprism' with the names of the base polygons, for examp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stericated 6-demicube
In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope. There are 8 pentic forms of the 6-cube. Pentic 6-cube The ''pentic 6-cube'', , has half of the vertices of a pentellated 6-cube, . Alternate names * Stericated 6-demicube/demihexeract * Small cellated hemihexeract (Acronym: sochax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±1,±1,±1,±3) with an odd number of plus signs. Images Penticantic 6-cube The ''penticantic 6-cube'', , has half of the vertices of a penticantellated 6-cube, . Alternate names * Steritruncated 6-demicube/demihexeract * cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±3,±3,±3,±5) with an odd number of plus signs. Images Pentiruncic 6-cube The ''pentiruncic 6-cube' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-cube Honeycomb
The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space. Constructions There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol . Another form has two alternating 6-cube facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol (6). Related honeycombs The ,34,4 , Coxeter group generates 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb. The ''6-cubic honeycomb'' can be alternated into the 6-demicubic honeycomb, replacing the 6-cubes with 6-demicubes, and the alternated gaps are filled by 6-orthoplex face ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-demicubic Honeycomb
The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb. It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h and the alternated vertices create 6-orthoplex facets. D6 lattice The vertex arrangement of the 6-demicubic honeycomb is the D6 lattice. The 60 vertices of the rectified 6-orthoplex vertex figure of the ''6-demicubic honeycomb'' reflect the kissing number 60 of this lattice. The best known is 72, from the E6 lattice and the 222 honeycomb. The D lattice (also called D) can be constructed by the union of two D6 lattices. This packing is only a lattice for even dimensions. The kissing number is 25=32 (2n-1 for n8). : ∪ The D lattice (also called D and C) can be constructed by the union of all four 6-demicubic lattices: It is also the 6-dimensional body centered cubic, the union of ... [...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] |
