6-simplex honeycomb

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6-simplex honeycomb
(No image)
Type Uniform 6-honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[7]} = 0[7]
Coxeter diagram CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
6-face types {35} 6-simplex t0.svg , t1{35} 6-simplex t1.svg
t2{35} 6-simplex t2.svg
5-face types {34} 5-simplex t0.svg , t1{34} 5-simplex t1.svg
t2{34} 5-simplex t2.svg
4-face types {33} 4-simplex t0.svg , t1{33} 4-simplex t1.svg
Cell types {3,3} 3-simplex t0.svg , t1{3,3} 3-simplex t1.svg
Face types {3} 2-simplex t0.svg
Vertex figure t0,5{35} 6-simplex t05.svg
Symmetry ×2, [[3[7]]]
Properties vertex-transitive

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.

Contents

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 Coxeter group. [1] 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*
6
lattice (also called A7
6
) 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.

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 10lr.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 01lr.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01lr.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 01l.png = dual of CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

This honeycomb is one of 17 unique uniform honeycombs [2] constructed by the Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

A6 honeycombs
Heptagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
a1[3[7]]CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png

CDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.png

i2[[3[7]]]CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel 3ab.pngCDel nodeab c3.pngCDel 3ab.pngCDel branch c4.png×2

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png 1 CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png 2 CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

r14[7[3[7]]]CDel node c1.pngCDel split1.pngCDel nodeab c1.pngCDel 3ab.pngCDel nodeab c1.pngCDel 3ab.pngCDel branch c1.png×14

CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png 3

Projection by folding

The 6-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

See also

Regular and uniform honeycombs in 6-space:

Notes

  1. "The Lattice A6". Archived from the original on 2012-01-19. Retrieved 2011-05-11.

Related Research Articles

In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.

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

The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.

The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic 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 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each 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.

<span class="mw-page-title-main">Cyclotruncated 5-simplex honeycomb</span>

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 simplicial 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.

<span class="mw-page-title-main">Simplicial honeycomb</span> Tiling of n-dimensional space

In geometry, the simplicial honeycomb is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It 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.

<span class="mw-page-title-main">Cyclotruncated simplicial honeycomb</span>

In geometry, the cyclotruncated simplicial 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 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.

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 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.

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

References

Space Family / /
E2 Uniform tiling 0[3] δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 hδ4 qδ4
E4 Uniform 4-honeycomb 0[5] δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 hδ6 qδ6
E6 Uniform 6-honeycomb 0[7] δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb 0[8] δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb 0[9] δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb 0[10]δ10hδ10qδ10
E10Uniform 10-honeycomb0[11]δ11hδ11qδ11
En-1Uniform (n-1)-honeycomb 0[n] δn hδn qδn 1k22k1k21