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Quarter 7-cubic Honeycomb
In seven-dimensional Euclidean geometry, the quarter 7-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 7-demicubic honeycomb, and a quarter of the vertices of a 7-cube honeycomb.Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318 Its facets are 7-demicubes, pentellated 7-demicubes, and × duoprisms. Related honeycombs See also Regular and uniform honeycombs in 7-space: * 7-cube honeycomb * 7-demicube honeycomb *7-simplex honeycomb * Truncated 7-simplex honeycomb *Omnitruncated 7-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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 7-honeycomb
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 ina ... [...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. 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 cemented ceramic squares or hexagons. Such t ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omnitruncated 7-simplex Honeycomb
In seven-dimensional Euclidean geometry, the omnitruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 7-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). A7* lattice The A lattice (also called A) is the union of eight A7 lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex. : ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of . Related polytopes and honeycombs See also Regular and uniform honeycombs in 7-space: *7-cubic honeycomb The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 7-space. It is analogous to the square tili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 7-simplex Honeycomb
In seven-dimensional Euclidean geometry, the cyclotruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, truncated 7-simplex, bitruncated 7-simplex, and tritruncated 7-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb. Structure It can be constructed by eight sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 6-simplex honeycomb divisions on each hyperplane. Related polytopes and honeycombs See also Regular and uniform honeycombs in 7-space: *7-cubic honeycomb * 7-demicubic honeycomb *7-simplex honeycomb *Omnitruncated 7-simplex honeycomb In seven-dimensional Euclidean geometry, the omnitruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 7-simplex facets. The facets of all omnitruncated simplectic honeycombs ... * 331 honeycomb Notes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-simplex Honeycomb
In seven-dimensional Euclidean geometry, the 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, rectified 7-simplex, birectified 7-simplex, and trirectified 7-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb. A7 lattice This vertex arrangement is called the A7 lattice or 7-simplex lattice. The 56 vertices of the expanded 7-simplex vertex figure represent the 56 roots of the _7 Coxeter group. It is the 7-dimensional case of a simplectic honeycomb. Around each vertex figure are 254 facets: 8+8 7-simplex, 28+28 rectified 7-simplex, 56+56 birectified 7-simplex, 70 trirectified 7-simplex, with the count distribution from the 9th row of Pascal's triangle. _7 contains _7 as a subgroup of index 144.N.W. Johnson: ''Geometries and Transformations'', (2018) 12.4: Euclidean Coxeter groups, p.294 Both _7 and _7 can be seen as affine extensions from A_7 from diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-demicube Honeycomb
The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb. It is composed of two different types of facets. The 7-cubes become alternated into 7-demicubes h and the alternated vertices create 7-orthoplex facets. D7 lattice The vertex arrangement of the 7-demicubic honeycomb is the D7 lattice. The 84 vertices of the rectified 7-orthoplex vertex figure of the ''7-demicubic honeycomb'' reflect the kissing number 84 of this lattice. The best known is 126, from the E7 lattice and the 331 honeycomb. The D packing (also called D) can be constructed by the union of two ''D7 lattices''. The D packings form lattices only in even dimensions. The kissing number is 26=64 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8). : ∪ The D lattice (also called D and C) can be constructed by the union of all four 7-demicubic lattices: It is also t ... [...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 exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentellated 7-demicube
In seven-dimensional geometry, a hexic 7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms. Hexic 7-cube Cartesian coordinates The Cartesian coordinates for the vertices of a hexic 7-cube centered at the origin are coordinate permutations: : (±1,±1,±1,±1,±1,±1,±3) with an odd number of plus signs. Images Hexicantic 7-cube Images Hexiruncic 7-cube Images Hexisteric 7-cube Images Hexipentic 7-cube Images Hexiruncicantic 7-cube Images Hexistericantic 7-cube Images Hexipenticantic 7-cube Images Hexisteriruncic 7-cube Images Hexipentiruncic 7-cube Images Hexipentisteric 7-cube Images Hexisteriruncicantic 7-cube Images Hexipentiruncicantic 7-cube Images Hexipentisteriruncic 7-cube Images Hexipentistericantic 7-cube Images Hexipentisteriruncicantic 7-cube Images Related polytopes This polytope is based on the 7-demicube, a part of a dimensional family of uniform polytopes c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-cube Honeycomb
The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space. There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol . Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol 7. Related honeycombs The ,35,4 , Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb. The ''7-cubic honeycomb'' can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets. Quadritr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-demicubic Honeycomb
The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb. It is composed of two different types of facets. The 7-cubes become alternated into 7-demicubes h and the alternated vertices create 7-orthoplex facets. D7 lattice The vertex arrangement of the 7-demicubic honeycomb is the D7 lattice. The 84 vertices of the rectified 7-orthoplex vertex figure of the ''7-demicubic honeycomb'' reflect the kissing number 84 of this lattice. The best known is 126, from the E7 lattice and the 331 honeycomb. The D packing (also called D) can be constructed by the union of two ''D7 lattices''. The D packings form lattices only in even dimensions. The kissing number is 26=64 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8). : ∪ The D lattice (also called D and C) can be constructed by the union of all four 7-demicubic lattices: It is also t ... [...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 particu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
