Omnitruncated 5-simplex honeycomb

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.

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	${\displaystyle {\tilde {A}}_{5}}$×12, [6[3^[6]]]
Properties	vertex-transitive

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 ${\displaystyle {\tilde {A}}_{5}}$ Coxeter group. The extended symmetry of the hexagonal diagram of the ${\displaystyle {\tilde {A}}_{5}}$ 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]]		${\displaystyle {\tilde {A}}_{5}}$
d2	<[3^[6]]>		${\displaystyle {\tilde {A}}_{5}}$×21	1, , , ,
p2	[[3^[6]]]		${\displaystyle {\tilde {A}}_{5}}$×22	2,
i4	[<[3^[6]]>]		${\displaystyle {\tilde {A}}_{5}}$×21×22	,
d6	<3[3^[6]]>		${\displaystyle {\tilde {A}}_{5}}$×61
r12	[6[3^[6]]]		${\displaystyle {\tilde {A}}_{5}}$×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:

${\displaystyle {\tilde {A}}_{5}}$
${\displaystyle {\tilde {C}}_{3}}$

See also

Regular and uniform honeycombs in 5-space:

• 5-cube honeycomb
• 5-demicube honeycomb
• 5-simplex honeycomb

Notes

1. mathworld: Necklace, OEIS sequence 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 2-9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E^2	Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
E^3	Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
E^4	Uniform 4-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E^5	Uniform 5-honeycomb	{3^[6]}	δ6	hδ6	qδ6
E^6	Uniform 6-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
E^7	Uniform 7-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
E^8	Uniform 8-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E^9	Uniform 9-honeycomb	{3^[10]}	δ10	hδ10	qδ10
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21