8-simplex honeycomb

8-simplex honeycomb
(No image)
Type	Uniform 8-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3[9]} = 0[9]
Coxeter diagram
6-face types	{37} , t1{37} t2{37} File:8-simplex t2.svg, t3{37} File:8-simplex t3.svg
6-face types	{36} File:7-simplex t0.svg, t1{36} File:7-simplex t1.svg t2{36} File:7-simplex t2.svg, t3{36} File:7-simplex t2.svg
6-face types	{35} File:6-simplex t0.svg, t1{35} File:6-simplex t1.svg t2{35} File:6-simplex t2.svg
5-face types	{34} File:5-simplex t0.svg, t1{34} File:5-simplex t1.svg t2{34} File:5-simplex t2.svg
4-face types	{33} File:4-simplex t0.svg, t1{33} File:4-simplex t1.svg
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 A³
₈ lattice is the union of three A₈ lattices, and also identical to the E8 lattice.

∪ File:CDel node.pngFile:CDel nodes 10lr.png ∪ File:CDel node.pngFile:CDel nodes 01lr.png = File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png.

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

∪ File:CDel node.pngFile:CDel nodes 10lr.png ∪ File:CDel node.pngFile:CDel nodes 01lr.png ∪ File:CDel node.pngFile:CDel nodes 10lr.png ∪ File:CDel node.pngFile:CDel nodes 01lr.png ∪ File:CDel node.pngFile:CDel nodes 10lr.png ∪ File:CDel node.pngFile:CDel nodes 01lr.png ∪ File:CDel node.pngFile:CDel branch 10l.png ∪ File:CDel node.pngFile:CDel branch 01l.png = dual of File:CDel nodes 11.pngFile:CDel nodes 11.pngFile:CDel nodes 11.pngFile:CDel branch 11.png.

Related polytopes and honeycombs

This honeycomb is one of 45 unique uniform honeycombs constructed by the ${\tilde{A}}_8$ 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:

${\tilde{A}}_8$
${\tilde{C}}_4$	File:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png

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

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, [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
Space	Family	${\tilde{A}}_{n-1}$	${\tilde{C}}_{n-1}$	${\tilde{B}}_{n-1}$	${\tilde{D}}_{n-1}$	${\tilde{G}}_2$ / ${\tilde{F}}_4$ / ${\tilde{E}}_{n-1}$
E2	Uniform tiling	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En−1	Uniform (n−1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21