Cyclotruncated 7-simplex honeycomb

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Cyclotruncated 7-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Cyclotruncated simplectic honeycomb
Schläfli symbol t0,1{3[8]}
Coxeter diagram CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
7-face types {36} 7-simplex t0.svg
t0,1{36} 7-simplex t01.svg
t1,2{36} 7-simplex t12.svg
t2,3{36} 7-simplex t23.svg
Vertex figureElongated 6-simplex antiprism
Symmetry ×22, [[3[8]]]
Properties vertex-transitive

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.

Euclidean geometry mathematical system attributed to Euclid

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.

Tessellation tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

Honeycomb (geometry) tiling of 3-or-more dimensional euclidian or hyperbolic space

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.

Contents

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.

Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. 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 machine learning, hyperplanes are a key tool to create support vector machines for such tasks as computer vision and natural language processing.

In six-dimensional Euclidean geometry, the cyclotruncated 6-simplex honeycomb is a space-filling tessellation. 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.

This honeycomb is one of 29 unique uniform honeycombs [1] constructed by the Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935.

See also

Regular and uniform honeycombs in 7-space:

The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation in Euclidean 7-space.

The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb.

In seven-dimensional Euclidean geometry, the 7-simplex honeycomb is a space-filling tessellation. 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.

Notes

  1. Weisstein, Eric W. "Necklace". MathWorld ., OEIS sequenceA000029 30-1 cases, skipping one with zero marks

Related Research Articles

Uniform 7-polytope vertex-transitive 7-polytope bounded by uniform facets

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.

Uniform 6-polytope vertex-transitive 6-polytope bounded by uniform facets

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

The 5-demicube honeycomb, or demipenteractic honeycomb is a uniform space-filling tessellation in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb.

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1.

In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation. Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.

Cyclotruncated 5-simplex honeycomb

In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation. It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1:1:1.

In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation. It is composed entirely of omnitruncated 5-simplex facets.

In geometry an omnitruncated simplectic honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry of the affine Coxeter group. Each is composed of omnitruncated simplex facets. The vertex figure for each is an irregular n-simplex.

In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation. 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.

Simplectic honeycomb dimensional infinite series of honeycombs

In geometry, the simplectic honeycomb is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is given a Schläfli symbol {3[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-simplex.

Cyclotruncated simplectic honeycomb

In geometry, the cyclotruncated simplectic honeycomb is a dimensional infinite series of honeycombs, based on the symmetry of the affine Coxeter group. It is given a Schläfli symbol t0,1{3[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with two adjacent nodes ringed. It is composed of n-simplex facets, along with all truncated n-simplices.

In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation. 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.

In six-dimensional Euclidean geometry, the omnitruncated 6-simplex honeycomb is a space-filling tessellation. It is composed entirely of omnitruncated 6-simplex facets.

In eight-dimensional Euclidean geometry, the omnitruncated 8-simplex honeycomb is a space-filling tessellation. It is composed entirely of omnitruncated 8-simplex facets.

In seven-dimensional Euclidean geometry, the omnitruncated 7-simplex honeycomb is a space-filling tessellation. It is composed entirely of omnitruncated 7-simplex facets.

In eight-dimensional Euclidean geometry, the cyclotruncated 8-simplex honeycomb is a space-filling tessellation. 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.

In seven-dimensional Euclidean geometry, the quarter 7-cubic honeycomb is a uniform space-filling tessellation. It has half the vertices of the 7-demicubic honeycomb, and a quarter of the vertices of a 7-cube honeycomb. Its facets are 7-demicubes, pentellated 7-demicubes, and {31,1,1}×{3,3} duoprisms.

References

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]}δ10hδ10qδ10
En-1Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21