Omnitruncated 8-simplex honeycomb

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.

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	${\displaystyle {\tilde {A}}_{9}}$×18, [9[3[9]]]
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).

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 ${\displaystyle {\tilde {A}}_{8}}$ 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]]		${\displaystyle {\tilde {A}}_{8}}$
i2	[[3[9]]]		${\displaystyle {\tilde {A}}_{8}}$×2	1 2
i6	[3[3[9]]]		${\displaystyle {\tilde {A}}_{8}}$×6
r18	[9[3[9]]]		${\displaystyle {\tilde {A}}_{8}}$×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