7-simplex honeycomb

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7-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3^[8]}
Coxeter–Dynkin diagram
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	t_0,6{3⁶}
Symmetry	${{\tilde {A}}}_{7}$ ×2, <[3^[8]]>
Properties	vertex-transitive

In seven-dimensional Euclidean geometry, the 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, rectified 7-simplex, birectified 7-simplex, and trirectified 7-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.

A7 lattice

This vertex arrangement is called the A7 lattice or 7-simplex lattice. The 56 vertices of the expanded 7-simplex vertex figure represent the 56 roots of the ${{\tilde {A}}}_{7}$ Coxeter group.^[1] It is the 7-dimensional case of a simplectic honeycomb. Around each vertex figure are 254 facets: 8+8 7-simplex, 28+28 rectified 7-simplex, 56+56 birectified 7-simplex, 70 trirectified 7-simplex, with the count distribution from the 9th row of Pascal's triangle.

${{\tilde {E}}}_{7}$ contains ${{\tilde {A}}}_{7}$ as a subgroup of index 144.^[2] Both ${{\tilde {E}}}_{7}$ and ${{\tilde {A}}}_{7}$ can be seen as affine extensions from $A_{7}$ from different nodes:

The A2
7 lattice can be constructed as the union of two A₇ lattices, and is identical to the E7 lattice.

+ = .

The A4
7 lattice is the union of four A₇ lattices, which is identical to the E7* lattice (or E2
7).

+ + + .

The A*
7 lattice (also called A8
7) is the union of eight A₇ lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.

+ + + + + + + = dual of .

Related polytopes and honeycombs

This honeycomb is one of 29 unique uniform honeycombs^[3] constructed by the ${{\tilde {A}}}_{7}$ Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:

Octagon symmetry	Extended symmetry	Extended order	Honeycombs
a1	[3^[8]]	×1
d2	<[3^[8]]>	×2	₁
p2	[[3^[8]]]	×2	₂
d4	<2[3^[8]]>	×4
p4	[2[3^[8]]]	×4
d8	[4[3^[8]]]	×8
r16	[8[3^[8]]]	×16	₃

Projection by folding

The 7-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:

${{\tilde {A}}}_{7}$
${{\tilde {C}}}_{4}$

Notes

↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A7.html
↑ N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 12: Euclidean symmetry groups, p 177
↑ Weisstein, Eric W., "Necklace", MathWorld., A000029 30-1 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
- (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]

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–11
Family	${{\tilde {A}}}_{{n-1}}$	${{\tilde {C}}}_{{n-1}}$	${{\tilde {B}}}_{{n-1}}$	${{\tilde {D}}}_{{n-1}}$	${{\tilde {G}}}_{2}$ / ${{\tilde {F}}}_{4}$ / ${{\tilde {E}}}_{{n-1}}$
Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
Uniform 5-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
Uniform 7-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
Uniform 8-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
Uniform 9-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
Uniform n-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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