Variable geometry turbomachine

8-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3[9]}
Coxeter–Dynkin diagrams	Template:CDD
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} File:3-simplex t0.svg, t1{3,3} File:3-simplex t1.svg
Face types	{3} File:2-simplex t0.svg
Vertex figure	t0,7{37} File:8-simplex t07.svg
Symmetry	${\tilde{A}}_8$×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 ${\tilde{A}}_8$ Coxeter group. 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.

${\tilde{E}}_8$ contains ${\tilde{A}}_8$ as a subgroup of index 5760. Both ${\tilde{E}}_8$ and ${\tilde{A}}_8$ can be seen as affine extensions of $A_8$ from different nodes: File:Affine A8 E8 relations.png

The ATemplate:Sup sub lattice is the union of three A₈ lattices, and also identical to the E8 lattice.

Template:CDD + Template:CDD + Template:CDD = Template:CDD.

The ATemplate:Sup sub lattice (also called ATemplate:Sup sub) 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

Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD = dual of Template:CDD.

Related polytopes and honeycombs

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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:

${\tilde{A}}_8$	Template:CDD
${\tilde{C}}_4$	Template:CDD

See also

• Regular and uniform honeycombs in 8-space:
  • 8-cubic honeycomb
  • 8-demicubic honeycomb
  • Truncated 8-simplex honeycomb
  • 5₂₁ honeycomb
  • 2₅₁ honeycomb
  • 1₅₂ honeycomb

Notes

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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]

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