6-simplex honeycomb

6-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3^[7]}
Coxeter diagram
6-face types	{3⁵} , t₁{3⁵} t₂{3⁵}
5-face types	{3⁴} , t₁{3⁴} t₂{3⁴}
4-face types	{3³} , t₁{3³}
Cell types	{3,3} , t₁{3,3}
Face types	{3}
Vertex figure	t_0,5{3⁵}
Symmetry	${\tilde{A}}_6$ ×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.

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 ${\tilde{A}}_6$ 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.

∪ ∪ ∪ ∪ ∪ ∪ = dual of

Related polytopes and honeycombs

This honeycomb is one of 17 unique uniform honeycombs^[2] constructed by the ${\tilde{A}}_6$ Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

A6 honeycombs

Heptagon symmetry	Extended symmetry	Extended group	Honeycombs
a1	[3^[7]]	${\tilde{A}}_6$
i2	[[3^[7]]]	${\tilde{A}}_6$ ×2	₁ ₂
r14	[7[3^[7]]]	${\tilde{A}}_6$ ×14	₃

Template:Hidden A6

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:

${\tilde{A}}_6$
${\tilde{C}}_3$

Notes

↑ http://www2.research.att.com/~njas/lattices/A6.html
↑
- Weisstein, Eric W., "Necklace", MathWorld., A000029 18-1 cases, skipping one with zero marks

References

Norman Johnson Uniform Polytopes, Manuscript (1991)
Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

Fundamental convex regular and uniform honeycombs in dimensions 2–10
Family	${\tilde{A}}_{n-1}$	${\tilde{C}}_{n-1}$	${\tilde{B}}_{n-1}$	${\tilde{D}}_{n-1}$	${\tilde{G}}_2$ / ${\tilde{F}}_4$ / ${\tilde{E}}_{n-1}$
Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
Uniform 5-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
Uniform 7-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
Uniform 8-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
Uniform 9-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
Uniform n-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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