8-simplex honeycomb

8-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3[9]}
Coxeter–Dynkin diagrams
6-face types	{37} t1{37} t2{37} t3{37}
6-face types	{36} t1{36} t2{36} t3{36}
6-face types	{35} t1{35} t2{35}
5-face types	{34} t1{34} t2{34}
4-face types	{33} t1{33}
Cell types	{3,3} t1{3,3}
Face types	{3}
Vertex figure	t0,7{37}
Symmetry	${\tilde {A}}_{8}$×2, <[3[9]]>
Properties	vertex-transitive

In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, rectified 8-simplex, birectified 8-simplex, and trirectified 8-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb.

A8 lattice

This vertex arrangement is called the A8 lattice or 8-simplex lattice. The 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the ${\tilde {A}}_{8}$ Coxeter group. It is the 8-dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9 8-simplex, 36+36 rectified 8-simplex, 84+84 birectified 8-simplex, 126+126 trirectified 8-simplex, with the count distribution from the 10th row of Pascal's triangle.

Related polytopes and honeycombs

This honeycomb is one of 45 unique uniform honycombs constructed by the ${\tilde {A}}_{8}$ Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter–Dynkin diagrams:

Symmetry	Order	Honeycombs
[3[9]]	Full
[[3[9]]]	×2
[3[3[9]]]	×6
[9[3[9]]]	×18

Projection by folding

The 8-simplex honeycomb can be projected into the 4-dimensional tesseractic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

${\tilde {A}}_{8}$
${\tilde {C}}_{4}$

The vertices of the [[5₂₁ honeycomb]] can also be expressed as a union of the vertices of three [8-simplex honeycombs:

= + +

See also

• Regular and uniform honeycombs in 8-space:
  • 8-cubic honeycomb
  • 8-demicubic honeycomb
  • 5₂₁ honeycomb
  • 2₅₁ honeycomb
  • 1₅₂ honeycomb

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 [1]
  • (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]