Omnitruncated 6-simplex honeycomb

Omnitruncated 6-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Omnitruncated simplectic honeycomb
Schläfli symbol	{3^[8]}
Coxeter–Dynkin diagrams
Facets	t0,1,2,3,4,5{3,3,3,3,3}
Vertex figure	Irr. 6-simplex
Symmetry	${\tilde {A}}_{7}$×14, [7[3^[7]]]
Properties	vertex-transitive

In six-dimensional Euclidean geometry, the omnitruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 6-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).

A*
6 lattice

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^[1] 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 diagram	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

Projection by folding

The omnitruncated 6-simplex honeycomb can be projected into the 4-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

${\tilde {A}}_{7}$
${{\tilde {C}}}_{4}$

See also

Regular and uniform honeycombs in 6-space:

• 6-cubic honeycomb
• 6-demicubic honeycomb
• 6-simplex honeycomb
• Truncated 6-simplex honeycomb
• 2₂₂ honeycomb

Notes

1. • 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 3–10 (or 2-9)
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 10-honeycomb	{3^[10]}	δ10	hδ10	qδ10
Uniform n-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21