In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling

tessellation A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

(or honeycomb). The tessellation fills space by

8-simplex In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is co ...

rectified 8-simplex In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a Rectification (geometry), rectification of the regular 8-simplex. There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rec ...

, 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

_8

Coxeter group. It is the 8-dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9

, 36+36

, 84+84 birectified 8-simplex, 126+126 trirectified 8-simplex, with the count distribution from the 10th row of Pascal's triangle.

_8

contains

_8

as a subgroup of index 5760. Both

_8

and

_8

can be seen as affine extensions of

A_8

from different nodes: The A lattice is the union of three A₈ lattices, and also identical to the

E8 lattice In mathematics, the E lattice is a special lattice in R. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E root system. The normIn th ...

.Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950) : ∪ ∪ = . The A lattice (also called A) 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 In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed t ...

of this lattice is an omnitruncated 8-simplex : ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .

Related polytopes and honeycombs

Projection by folding

The ''8-simplex honeycomb'' can be projected into the 4-dimensional

tesseractic honeycomb In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensional packing of tesseract facets. Its verte ...

by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

Notes

References

* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) * Kaleidoscopes: Selected Writings of

H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. 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'', ath. Zeit. 200 (1988) 3–45 {{Honeycombs Honeycombs (geometry) 9-polytopes

A8 lattice

Related polytopes and honeycombs

Projection by folding

See also

Notes

References