Omnitruncated 8-simplex honeycomb

Omnitruncated 8-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Omnitruncated simplectic honeycomb
Schläfli symbol	{3^[9]}
Coxeter–Dynkin diagrams
7-face types	t01234567{3,3,3,3,3,3,3}
Vertex figure	Irr. 8-simplex
Symmetry	${\tilde{A}}_9$×18, [9[3^[9]]]
Properties	vertex-transitive

In eight-dimensional Euclidean geometry, the omnitruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 8-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*
8 lattice

The A*
8 lattice (also called A9
8) is the union of nine A₈ lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex

∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .

Related polytopes and honeycombs

This honeycomb is one of 45 unique uniform honeycombs^[1] constructed by the ${\tilde{A}}_8$ Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:

A8 honeycombs
Enneagon symmetry	Symmetry	Extended diagram	Extended group	Honeycombs
a1	[3^[9]]		${\tilde{A}}_8$
i2	[[3^[9]]]		${\tilde{A}}_8$×2	1 2
i6	[3[3^[9]]]		${\tilde{A}}_8$×6
r18	[9[3^[9]]]		${\tilde{A}}_8$×18	3

See also

Regular and uniform honeycombs in 8-space:

• 8-cubic honeycomb
• 8-demicubic honeycomb
• 8-simplex honeycomb
• Truncated 8-simplex honeycomb
• 5₂₁ honeycomb
• 2₅₁ honeycomb
• 1₅₂ honeycomb

Notes

1. • Weisstein, Eric W., "Necklace", MathWorld., A000029 46-1 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]