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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] |
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] |
Seventh Dimension
In mathematics, a sequence of ''n'' real numbers can be understood as a location in ''n''-dimensional space. When ''n'' = 7, the set of all such locations is called 7-dimensional space. Often such a space is studied as a vector space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equipped with a Euclidean metric, which is defined by the dot product. More generally, the term may refer to a seven-dimensional vector space over any field, such as a seven-dimensional complex vector space, which has 14 real dimensions. It may also refer to a seven-dimensional manifold such as a 7-sphere, or a variety of other geometric constructions. Seven-dimensional spaces have a number of special properties, many of them related to the octonions. An especially distinctive property is that a cross product can be defined only in three or seven dimensions. This is related to Hurwitz's theorem, which prohibits the existence of algebraic structures l ... [...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] |
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 face ... [...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] |
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 ... [...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] |
Cyclotruncated 6-simplex Honeycomb
In six-dimensional Euclidean geometry, the cyclotruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb. Structure It can be constructed by seven sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-simplex honeycomb divisions on each hyperplane. Related polytopes and honeycombs See also Regular and uniform honeycombs in 6-space: * 6-cubic honeycomb * 6-demicubic honeycomb *6-simplex honeycomb In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportion ... * Omnitruncated 6-simplex honeycomb * 222 honeycomb Not ... [...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] |
Truncated 7-simplex
In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex. There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex. Truncated 7-simplex In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex. Alternate names * Truncated octaexon (Acronym: toc) (Jonathan Bowers) Coordinates The vertices of the ''truncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 8-orthoplex. Images Bitruncated 7-simplex Alternate names * Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers ... [...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] |
