Omnitruncated 5-simplex honeycomb

Omnitruncated 5-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Omnitruncated simplectic honeycomb
Schläfli symbol	t₀₁₂₃₄₅{3^[6]}
Coxeter–Dynkin diagram
5-face types	t₀₁₂₃₄{3,3,3,3}
4-face types	t₀₁₂₃{3,3,3} {}×t₀₁₂{3,3} {6}×{6}
Cell types	t₀₁₂{3,3} {4,3} {}x{6}
Face types	{4} {6}
Vertex figure	Irr. 5-simplex
Symmetry	${\tilde{A}}_5$ ×12, [6[3^[6]]]
Properties	vertex-transitive

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

A₅^* lattice

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^[1] 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 diagram	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 omnitruncated 5-simplex honeycomb can be projected into the 3-dimensional omnitruncated cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same 3-space vertex arrangement:

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

Notes

↑ , 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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