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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] |
Uniform 7-polytope
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose facets are uniform 6-polytopes. Regular 7-polytopes Regular 7-polytopes are represented by the Schläfli symbol with u 6-polytopes facets around each 4-face. There are exactly three such convex regular 7-polytopes: # - 7-simplex # - 7-cube # - 7-orthoplex There are no nonconvex regular 7-polytopes. Characteristics The topology of any given 7-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, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably disting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sixth Dimension
Six-dimensional space is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature. Formally, six-dimensional Euclidean space, ℝ6, is generated by considering all real 6-tuples as 6- vectors in this space. As such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined and can be used to calculate the metric. 6 × 6 matrices can be used to describe transformations such as rotations that keep the origin fixed. ... [...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] |
2 22 Honeycomb
In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol . It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex. Its vertex arrangement is the '' E6 lattice'', and the root system of the E6 Lie group so it can also be called the E6 honeycomb. Construction It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 6-dimensional space. The facet information can be extracted from its Coxeter–Dynkin diagram, . Removing a node on the end of one of the 2-node branches leaves the 221, its only facet type, The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 122, . The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, ''t''2, . The face figure is the vertex figure of the edge figure, here being a triangular duoprism, &ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omnitruncated 6-simplex Honeycomb
In six-dimensional Euclidean geometry, the omnitruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 6-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 seven A6 lattices, and has the vertex arrangement of the dual to the ''omnitruncated 6-simplex honeycomb'', and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex. : ∪ ∪ ∪ ∪ ∪ ∪ = dual of 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, rec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 proportions of 1:1:1 respectively in the whole honeycomb. A6 lattice This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the _6 Coxeter group. It is the 6-dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6-simplex, 21+21 rectified 6-simplex, 35+35 birectified 6-simplex, with the count distribution from the 8th row of Pascal's triangle. The A lattice (also called A) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex. : ∪ ∪ ∪ ∪ ∪ ∪ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-demicubic Honeycomb
The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb. It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h and the alternated vertices create 6-orthoplex facets. D6 lattice The vertex arrangement of the 6-demicubic honeycomb is the D6 lattice. The 60 vertices of the rectified 6-orthoplex vertex figure of the ''6-demicubic honeycomb'' reflect the kissing number 60 of this lattice. The best known is 72, from the E6 lattice and the 222 honeycomb. The D lattice (also called D) can be constructed by the union of two D6 lattices. This packing is only a lattice for even dimensions. The kissing number is 25=32 (2n-1 for n8). : ∪ The D lattice (also called D and C) can be constructed by the union of all four 6-demicubic lattices: It is also the 6-dimensional body centered cubic, the union o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-cubic Honeycomb
The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space. Constructions There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol . Another form has two alternating 6-cube facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol (6). Related honeycombs The ,34,4 , Coxeter group generates 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb. The ''6-cubic honeycomb'' can be alternated into the 6-demicubic honeycomb, replacing the 6-cubes with 6-demicubes, and the alternated gaps are filled by 6-orth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclotruncated 5-simplex Honeycomb
In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1:1:1. Structure Its vertex figure is an elongated 5-cell antiprism, two parallel 5-cells in dual configurations, connected by 10 tetrahedral pyramids (elongated 5-cells) from the cell of one side to a point on the other. The vertex figure has 8 vertices and 12 5-cells. It can be constructed as six sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-cell honeycomb divisions on each hyperplane. Related polytopes and honeycombs See also Regular and uniform honeycombs in 5-space: *5-cubic honeycomb * 5-demicubic honeycomb * 5-simplex honeycomb In Five-dimensional space, five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-fil ... [...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''-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 6-simplex
In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the tetrahedral cells of the 6-simplex. Truncated 6-simplex Alternate names * Truncated heptapeton (Acronym: til) (Jonathan Bowers) Coordinates The vertices of the ''truncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex. Images Bitruncated 6-simplex Alternate names * Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers) Coordinates The vertices of the ''bitruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0 ... [...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] |
