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Quarter 8-cubic Honeycomb
In seven-dimensional Euclidean geometry, the quarter 8-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 8-demicubic honeycomb, and a quarter of the vertices of a 8-cube honeycomb.Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318 Its facets are 8-demicubes h, pentic 8-cubes h6, × and × duoprisms. See also Regular and uniform honeycombs in 8-space: * 8-cube honeycomb *8-demicube honeycomb *8-simplex honeycomb *Truncated 8-simplex honeycomb * Omnitruncated 8-simplex honeycomb Notes References * Kaleidoscopes: Selected Writings of H. S. M. 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 t ..., edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 9-polytope
In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets. A uniform 9-polytope is one which is vertex-transitive, and constructed from uniform 8-polytope Facet (geometry), facets. Regular 9-polytopes Regular 9-polytopes can be represented by the Schläfli symbol , with w 8-polytope facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ... around each Peak (geometry), peak. There are exactly three such List of regular polytopes#Convex 4, convex regular 9-polytopes: # - 9-simplex # - 9-cube # - 9-orthoplex There are no nonconvex regular 9-polytopes. Euler characteristic The topology of any given 9-polytope is defined by its Betti numbers and tor ... [...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 8-simplex Honeycomb
In eight-dimensional Euclidean geometry, the omnitruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 8-simplex facets. The facets of all omnitruncated simplectic honeycombs are called 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 nine A8 lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex : ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of . Related polytopes and honeycombs See also Regular and uniform honeycombs in 8-space: * 8-cubic honeycomb * 8-demicubic honeycomb *8-simplex honeycomb *Truncated 8-simplex honeycomb In Eighth dimension, eight-dimensional Euclidean geometry, the cyclotruncated 8-simplex honeycomb is a space-fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 8-simplex Honeycomb
In Eighth dimension, eight-dimensional Euclidean geometry, the cyclotruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb (geometry), honeycomb). The tessellation fills space by 8-simplex, truncated 8-simplex, bitruncated 8-simplex, tritruncated 8-simplex, and quadritruncated 8-simplex facets. These facet types occur in proportions of 2:2:2:2:1 respectively in the whole honeycomb. Structure It can be constructed by nine sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 7-simplex honeycomb divisions on each hyperplane. Related polytopes and honeycombs See also Regular and uniform honeycombs in 8-space: *8-cubic honeycomb *8-demicubic honeycomb *8-simplex honeycomb *Omnitruncated 8-simplex honeycomb *5 21 honeycomb, 521 honeycomb *2 51 honeycomb, 251 honeycomb *1 52 honeycomb, 152 honeycomb Notes References * Norman Johnson (mathematician), Norman Johnson ''Uniform Polytopes'', Manuscript (1991) * Kal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-simplex Honeycomb
In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, rectified 8-simplex, birectified 8-simplex, and trirectified 8-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb. A8 lattice This vertex arrangement is called the A8 lattice or 8-simplex lattice. The 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the _8 Coxeter group. It is the 8-dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9 8-simplex, 36+36 rectified 8-simplex, 84+84 birectified 8-simplex, 126+126 trirectified 8-simplex, with the count distribution from the 10th row of Pascal's triangle. _8 contains _8 as a subgroup of index 5760. Both _8 and _8 can be seen as affine extensions of A_8 from different nodes: The A lattice is the union of three A8 lattices, and also identical to the E8 lattice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-demicube Honeycomb
The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb. It is composed of two different types of facets. The 8-cubes become alternated into 8-demicubes h and the alternated vertices create 8-orthoplex facets . D8 lattice The vertex arrangement of the 8-demicubic honeycomb is the D8 lattice. The 112 vertices of the rectified 8-orthoplex vertex figure of the ''8-demicubic honeycomb'' reflect the kissing number 112 of this lattice. The best known is 240, from the E8 lattice and the 521 honeycomb. _8 contains _8 as a subgroup of index 270. Both _8 and _8 can be seen as affine extensions of D_8 from different nodes: The D lattice (also called D) can be constructed by the union of two D8 lattices. This packing is only a lattice for even dimensions. The kissing number is 240. (2n-1 for n8). It is identical to the E8 lattice. At 8-dime ... [...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] |
Pentic 8-cube
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets. A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets. Regular 8-polytopes Regular 8-polytopes can be represented by the Schläfli symbol , with v 7-polytope facets around each peak. There are exactly three such convex regular 8-polytopes: # - 8-simplex # - 8-cube # - 8-orthoplex There are no nonconvex regular 8-polytopes. Characteristics The topology of any given 8-polytope is defined by its Betti numbers and 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 characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8-polytopes, whatever their underlying topology. This inadequacy of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-cube Honeycomb
The 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 8-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space. There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol . Another form has two alternating hypercube facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 256 types of facets around each vertex and a prismatic product Schläfli symbol 8. Related honeycombs The ,36,4 , Coxeter group generates 511 permutations of uniform tessellations, 271 with unique symmetry and 270 with unique geometry. The expanded 8-cubic honeycomb is geometrically identical to the 8-cubic honeycomb. The ''8-cubic honeycomb'' can be alternated into the 8-demicubic honeycomb, replacing the 8-cubes with 8-demicubes, and the alter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-demicubic Honeycomb
The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb. It is composed of two different types of facets. The 8-cubes become alternated into 8-demicubes h and the alternated vertices create 8-orthoplex facets . D8 lattice The vertex arrangement of the 8-demicubic honeycomb is the D8 lattice. The 112 vertices of the rectified 8-orthoplex vertex figure of the ''8-demicubic honeycomb'' reflect the kissing number 112 of this lattice. The best known is 240, from the E8 lattice and the 521 honeycomb. _8 contains _8 as a subgroup of index 270. Both _8 and _8 can be seen as affine extensions of D_8 from different nodes: The D lattice (also called D) can be constructed by the union of two D8 lattices. This packing is only a lattice for even dimensions. The kissing number is 240. (2n-1 for n8). It is identical to the E8 lattice. ... [...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] |
