8-simplex honeycomb

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.

8-simplex honeycomb
(No image)
TypeUniform 8-honeycomb
FamilySimplectic honeycomb
Schläfli symbol{3[9]}
Coxeter diagram
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}
5-face types{34} , t1{34}
4-face types{33} , t1{33}
Cell types{3,3} , t1{3,3}
Face types{3}
Vertex figuret0,7{37}
Symmetry×2, [[3[9]]]

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 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.

contains as a subgroup of index 5760.[2] Both and can be seen as affine extensions of from different nodes:

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

= .

The A*
lattice (also called A9
) 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 .

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:

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:

See also


  1. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A8.html
  2. N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p.294
  3. Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)


  • 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 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21
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