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]

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