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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] |
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] |
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. From there, he accepted a position in the Mathematics Department of Wheaton College in Massachusetts and taught until his retirement in 1998. Career In 1966, he enumerated 92 convex non-uniform polyhedra with regular faces. Victor Zalgaller later proved (1969) that Johnson's list was complete, and the set is now known a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 52 Honeycomb
In geometry, the 152 honeycomb is a Uniform honeycomb, uniform tessellation of 8-dimensional Euclidean space. It contains Gosset 1 42 polytope, 142 and demiocteract, 151 Facet (geometry), facets, in a birectified 8-simplex vertex figure. It is the final figure in the uniform 1 k2 polytope, 1k2 polytope family. Construction It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. : Removing the node on the end of the 2-length branch leaves the 8-demicube, 151. : Removing the node on the end of the 5-length branch leaves the 1 42 polytope, 142. : The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 8-simplex, 052. : Related polytopes and honeycombs See also * 5 21 honeycomb, 521 honeycomb * 2 51 honeycomb, 251 honeycomb References * Harold Scott MacDonald Coxeter, Coxeter ''The Beauty of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2 51 Honeycomb
In 8-dimensional geometry, the 251 honeycomb is a space-filling Uniform honeycomb, uniform tessellation. It is composed of Gosset 2 41 polytope, 241 polytope and 8-simplex Facet (geometry), facets arranged in an 8-demicube vertex figure. It is the final figure in the uniform 2 k1 polytope, 2k1 family. Construction It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. : Removing the node on the short branch leaves the 8-simplex. : Removing the node on the end of the 5-length branch leaves the Gosset 2 41 polytope, 241. : The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 8-demicube, 151. : The edge figure is the vertex figure of the vertex figure. This makes the rectified 7-simplex, 051. : Related polytopes and honeycombs References * Harold Scott MacDonald Coxeter, Coxeter ''The Beauty of G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5 21 Honeycomb
In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.Coxeter, 1973, Chapter 5: The Kaleidoscope By putting spheres at its vertices one obtains the densest-possible packing of spheres in 8 dimensions. This was proven by Maryna Viazovska in 2016 using the theory of modular forms. Viazovska was awarded the Fields Medal for this work in 2022. This honeycomb was first studied by Gosset who called it a ''9-ic semi-regular figure'' (Gosset regarded honeycombs in ''n'' dimensions as degenerate ''n''+1 polytopes). Each vertex of the 521 honeycomb is surrounded by 2160 8-orthoplexes and 17280 8-simplicies. The vertex figure of Gosset's honeycomb is the semiregular 421 polytope. It is the final figure in the k21 family. This honeycomb is highly regular in the sense that its symmetry group (the affine _8 Weyl group) acts transitively on the ''k''- ... [...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-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] |
8-cubic 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 altern ... [...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 In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other wo .... 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
