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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] |
Uniform 8-polytope
In Eight-dimensional space, eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope Ridge (geometry), ridge being shared by exactly two 7-polytope Facet (mathematics), 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 Facet (mathematics), facets around each Peak (geometry), peak. There are exactly three such List of regular polytopes#Convex 4, 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 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 u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eighth Dimension
In mathematics, a sequence of ''n'' real numbers can be understood as a location in ''n''-dimensional space. When ''n'' = 8, the set of all such locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance. Eight-dimensional Euclidean space is eight-dimensional space equipped with the Euclidean metric. More generally the term may refer to an eight-dimensional vector space over any field, such as an eight-dimensional complex vector space, which has 16 real dimensions. It may also refer to an eight-dimensional manifold such as an 8-sphere, or a variety of other geometric constructions. Geometry 8-polytope A polytope in eight dimensions is called an 8-polytope. The most studied are the regular polytopes, of which there are only three in eight dimensions: the 8-simplex, 8-cube, and 8-orthoplex. A broader family are the uniform 8-polytopes, constructed from fundamental symmetry domains of reflection, each domain ... [...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] |
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] |
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] |
Cyclotruncated 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 are ... * 331 honeycomb N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, hyperplanes may have different properties. For instance, a hyperplane of an -dimensional affine space is a flat subset with dimension and it separates the space into two half spaces. While a hyperplane of an -dimensional projective space does not have this property. The difference in dimension between a subspace and its ambient space is known as the codimension of with respect to . Therefore, a necessary and sufficient condition for to be a hyperplane in is for to have codimension one in . Technical description In geometry, a hyperplane of an ''n''-dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadritruncated 8-simplex
In eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex. There are four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the tetrahedral cells of the 8-simplex. Truncated 8-simplex Alternate names * Truncated enneazetton (Acronym: tene) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''truncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 9-orthoplex. Images Bitruncated 8-simplex Alternate names * Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''bitruncated 8-simplex'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 8-simplex
In eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex. There are four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the tetrahedral cells of the 8-simplex. Truncated 8-simplex Alternate names * Truncated enneazetton (Acronym: tene) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''truncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 9-orthoplex. Images Bitruncated 8-simplex Alternate names * Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''bitruncated 8-simplex'' can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
