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

(No image)  
Type  Uniform 8honeycomb 
Family  Simplectic honeycomb 
Schläfli symbol  {3^{[9]}} 
Coxeter diagram  
6face types  {3^{7}} t_{2}{3^{7}} 
6face types  {3^{6}} t_{2}{3^{6}} 
6face types  {3^{5}} t_{2}{3^{5}} 
5face types  {3^{4}} t_{2}{3^{4}} 
4face types  {3^{3}} 
Cell types  {3,3} 
Face types  {3} 
Vertex figure  t_{0,7}{3^{7}} 
Symmetry  ×2, [[3^{[9]}]] 
Properties  vertextransitive 
A8 lattice
This vertex arrangement is called the A8 lattice or 8simplex lattice. The 72 vertices of the expanded 8simplex vertex figure represent the 72 roots of the Coxeter group.[1] It is the 8dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9 8simplex, 36+36 rectified 8simplex, 84+84 birectified 8simplex, 126+126 trirectified 8simplex, with the count distribution from the 10th row of Pascal's triangle.
contains as a subgroup of index 5760.[2] Both and can be seen as affine extensions of from different nodes:
The A^{3}
_{8} lattice is the union of three A_{8} lattices, and also identical to the E8 lattice.[3]
∪ ∪ = .
The A^{*}
_{8} lattice (also called A^{9}
_{8}) is the union of nine A_{8} lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8simplex
Related polytopes and honeycombs
This honeycomb is one of 45 unique uniform honeycombs[4] constructed by the 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]}] 
 
i2  [[3^{[9]}]]  ×2 
 
i6  [3[3^{[9]}]]  ×6  
r18  [9[3^{[9]}]]  ×18 
Projection by folding
The 8simplex honeycomb can be projected into the 4dimensional tesseractic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
See also
 Regular and uniform honeycombs in 8space:
Notes
 http://www.math.rwthaachen.de/~Gabriele.Nebe/LATTICES/A8.html
 N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p.294
 Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)

 Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 461 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, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform spacefillings)
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
Fundamental convex regular and uniform honeycombs in dimensions 29  

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 