5-simplex honeycomb

5-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3^[6]}
Coxeter diagram
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_0,4{3⁴}
Coxeter groups	${\tilde{A}}_5$ ×2, <[3^[6]]>
Properties	vertex-transitive

In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). 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.

A5 lattice

This vertex arrangement is called the A₅ lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the ${\tilde{A}}_5$ Coxeter group.^[1] It is the 5-dimensional case of a simplectic honeycomb.

The A2
5 lattice is the union of two A₅ lattices:

∪

The A3
5 is the union of three A₅ lattices:

∪ ∪ .

The A*
5 lattice (also called A6
5) is the union of six A₅ lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.

∪ ∪ ∪ ∪ ∪ = dual of

Related polytopes and honeycombs

This honeycomb is one of 12 unique uniform honeycombs^[2] constructed by the ${\tilde{A}}_5$ Coxeter group. The extended symmetry of the hexagonal diagram of the ${\tilde{A}}_5$ Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:

A5 honeycombs

Hexagon symmetry	Extended symmetry	Extended group	Honeycomb diagrams
a1	[3^[6]]	${\tilde{A}}_5$
d2	<[3^[6]]>	${\tilde{A}}_5$ ×2₁	₁, , , ,
p2	[[3^[6]]]	${\tilde{A}}_5$ ×2₂	₂,
i4	[<[3^[6]]>]	${\tilde{A}}_5$ ×2₁×2₂	,
d6	<3[3^[6]]>	${\tilde{A}}_5$ ×6₁
r12	[6[3^[6]]]	${\tilde{A}}_5$ ×12	₃

Projection by folding

The 5-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:

${\tilde{A}}_5$
${\tilde{C}}_3$

Notes

↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A5.html
↑ , A000029 13-1 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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