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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] |
Uniform 8-polytope
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 inadeq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omnitruncated Simplectic Honeycomb
In geometry an omnitruncated simplectic honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry of the _n affine Coxeter group. Each is composed of omnitruncated simplex facets. The vertex figure for each is an irregular n-simplex. The facets of an ''omnitruncated simplectic honeycomb'' are called permutahedra and can be positioned in ''n+1'' space with integral coordinates, permutations of the whole numbers (0,1,..,n). Projection by folding The (2n-1)-simplex honeycombs can be projected into the n-dimensional omnitruncated hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement: See also * Hypercubic honeycomb * Alternated hypercubic honeycomb * Quarter hypercubic honeycomb * Simplectic honeycomb * Truncated simplectic honeycomb References * George Olshevsky, ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Johnson (mathematician)
Norman Woodason Johnson () was a mathematician at Wheaton College, Norton, Massachusetts. Early life and education Norman Johnson was born on in Chicago. His father had a bookstore and published a local newspaper. Johnson earned his undergraduate mathematics degree in 1953 at Carleton College in Northfield, Minnesota followed by a master's degree from the University of Pittsburgh. After graduating in 1953, Johnson did alternative civilian service as a conscientious objector. He earned his PhD from the University of Toronto in 1966 with a dissertation title of ''The Theory of Uniform Polytopes and Honeycombs'' under the supervision 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 .... From there, he accepted a position in the Mathematics Department of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3 31 Honeycomb
In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex. Construction It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. : Removing the node on the short branch leaves the 6-simplex facet: : Removing the node on the end of the 3-length branch leaves the 321 facet: : The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 231 polytope. : The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (131). : The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (031). : The cell figure is determined by removing the ringed node of the fac ... [...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-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] |
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 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 polytope, 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 [4,35,4], , Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The Expansion (geometry), expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb. The ''7-cubic honeycomb'' can be Alternation (geometry), alternated into the 7-demicubic honeycomb, replacing the 7-cubes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voronoi Cell
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer to that seed than to any other. The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation. The Voronoi diagram is named after mathematician Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons. Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art. The simplest case In the simplest case, shown in the first picture, we are given a finite set of points in the Euc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equal distance and angles from a center point. Two polytopes share the same ''vertex arrangement'' if they share the same 0-skeleton. A group of polytopes that shares a vertex arrangement is called an ''army''. Vertex arrangement The same set of vertices can be connected by edges in different ways. For example, the ''pentagon'' and ''pentagram'' have the same ''vertex arrangement'', while the second connects alternate vertices. A ''vertex arrangement'' is often described by the convex hull polytope which contains it. For example, the regular ''pentagram'' can be said to have a (regular) ''pentagonal vertex arrangement''. Infinite tilings can also share common ''vertex arrangements''. For example, this triangular lattice of points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A7 Lattice
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 differen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutahedron
In mathematics, the permutohedron of order ''n'' is an (''n'' − 1)-dimensional polytope embedded in an ''n''-dimensional space. Its vertex coordinates (labels) are the permutations of the first ''n'' natural numbers. The edges identify the shortest possible paths (sets of transpositions) that connect two vertices (permutations). Two permutations connected by an edge differ in only two places (one transposition), and the numbers on these places are neighbors (differ in value by 1). The image on the right shows the permutohedron of order 4, which is the truncated octahedron. Its vertices are the 24 permutations of (1, 2, 3, 4). Parallel edges have the same edge color. The 6 edge colors correspond to the 6 possible transpositions of 4 elements, i.e. they indicate in which two places the connected permutations differ. (E.g. red edges connect permutations that differ in the last two places.) History According to , permutohedra were first studied by . The name ''permu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
