8-simplex honeycomb

8-simplex honeycomb
(No image)
Type Uniform 8-honeycomb
Family Simplectic 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}
t2{35}
5-face types {34} , t1{34}
t2{34}
4-face types {33} , t1{33}
Cell types {3,3} , t1{3,3}
Face types {3}
Vertex figure t0,7{37}
Symmetry ${\displaystyle {\tilde {A}}_{8}}$×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 ${\displaystyle {\tilde {A}}_{8}}$  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.

${\displaystyle {\tilde {E}}_{8}}$  contains ${\displaystyle {\tilde {A}}_{8}}$  as a subgroup of index 5760.[2] Both ${\displaystyle {\tilde {E}}_{8}}$  and ${\displaystyle {\tilde {A}}_{8}}$  can be seen as affine extensions of ${\displaystyle A_{8}}$  from different nodes:

The A3
8
lattice is the union of three A8 lattices, and also identical to the E8 lattice.[3]

=                .

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[4] constructed by the ${\displaystyle {\tilde {A}}_{8}}$  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:

• Regular and uniform honeycombs in 8-space:

Notes

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)
4. ^ * Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence 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 [1]
• (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
${\displaystyle {\tilde {A}}_{n-1}}$  ${\displaystyle {\tilde {C}}_{n-1}}$  ${\displaystyle {\tilde {B}}_{n-1}}$  ${\displaystyle {\tilde {D}}_{n-1}}$  ${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21