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]

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
Uniform 5-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ6	hδ6	qδ6
Uniform 7-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
Uniform 8-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
Uniform 9-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
Uniform 10-honeycomb	{3^[10]}	δ10	hδ10	qδ10
Uniform n-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21