Omnitruncated 6-simplex honeycomb

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.

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	${\displaystyle {\tilde {A}}_{7}}$×14, [7[3[7]]]
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* 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 ${\displaystyle {\tilde {A}}_{6}}$ 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]]		${\displaystyle {\tilde {A}}_{6}}$
i2	[[3[7]]]		${\displaystyle {\tilde {A}}_{6}}$×2	1 2
r14	[7[3[7]]]		${\displaystyle {\tilde {A}}_{6}}$×14	3

See also

Regular and uniform honeycombs in 6-space:

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