Cyclotruncated 8-simplex honeycomb

In eight-dimensional Euclidean geometry, the cyclotruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, truncated 8-simplex, bitruncated 8-simplex, tritruncated 8-simplex, and quadritruncated 8-simplex facets. These facet types occur in proportions of 2:2:2:2:1 respectively in the whole honeycomb.

Cyclotruncated 8-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Cyclotruncated simplectic honeycomb
Schläfli symbol	t0,1{3[9]}
Coxeter diagram
8-face types	{37} , t0,1{37} t1,2{37} , t2,3{37} t3,4{37}
Vertex figure	Elongated 7-simplex antiprism
Symmetry	${\displaystyle {\tilde {A}}_{8}}$×2, [[3[9]]]
Properties	vertex-transitive

Structure

It can be constructed by nine sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 7-simplex honeycomb divisions on each hyperplane.

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
• Omnitruncated 8-simplex honeycomb
• 5₂₁ honeycomb
• 2₅₁ honeycomb
• 1₅₂ honeycomb