8-simplex honeycomb

8-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3^[9]}
Coxeter diagram
6-face types	{3^7} , t1{3^7} t2{3^7} , t3{3^7}
6-face types	{3^6} , t1{3^6} t2{3^6} , t3{3^6}
6-face types	{3^5} , t1{3^5} t2{3^5}
5-face types	{3^4} , t1{3^4} t2{3^4}
4-face types	{3^3} , t1{3^3}
Cell types	{3,3} , t1{3,3}
Face types	{3}
Vertex figure	t0,7{3^7}
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.^[1] 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.

${\tilde{E}}_8$ contains ${\tilde{A}}_8$ as a subgroup of index 5760.^[2] Both ${\tilde{E}}_8$ and ${\tilde{A}}_8$ can be seen as affine extensions of $A_8$ from different nodes:

The A3
8 lattice is the union of three A₈ lattices, and also identical to the E8 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^[3] 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

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:

${\tilde{A}}_8$
${\tilde{C}}_4$

See also

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

Notes

1. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A8.html
2. N.W. Johnson: Geometries and Transformations, (2015) Chapter 12: Euclidean symmetry groups, p.177
3. • 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]