In eighth-dimensional

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, 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 A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic Beeswax, wax cells built by honey bees in their beehive, nests to contain their larvae and stores of honey and pollen. beekeeping, Beekee ...

). 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 In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equ ...

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 In geometry, the simplectic honeycomb (or -simplex honeycomb) is a dimensional infinite series of honeycombs, based on the _n affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of nodes with one node ...

. 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 In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...

_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

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

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