7-simplex honeycomb

7-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3^[8]}
Coxeter diagram
6-face types	{3^6} , t1{3^6} t2{3^6} , t3{3^6}
6-face types	{3^5} , t1{3^5} t2{3^5}
5-face types	{3^4} , t1{3^4} t2{3^4}
4-face types	{3^3} , t1{3^3}
Cell types	{3,3} , t1{3,3}
Face types	{3}
Vertex figure	t0,6{3^6}
Symmetry	${\tilde{A}}_7$×21, <[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).

∪ ∪ ∪ = + = dual of .

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:

A7 honeycombs 

Octagon symmetry	Extended symmetry	Extended diagram	Extended group	Honeycombs
a1	[3^[8]]		${\tilde{A}}_7$
d2	<[3^[8]]>		${\tilde{A}}_7$×21	1
p2	[[3^[8]]]		${\tilde{A}}_7$×22	2
d4	<2[3^[8]]>		${\tilde{A}}_7$×41
p4	[2[3^[8]]]		${\tilde{A}}_7$×42
d8	[4[3^[8]]]		${\tilde{A}}_7$×8
r16	[8[3^[8]]]		${\tilde{A}}_7$×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:

${\tilde{A}}_7$
${\tilde{C}}_4$

See also

Regular and uniform honeycombs in 7-space:

• 7-cubic honeycomb
• 7-demicubic honeycomb
• Truncated 7-simplex honeycomb
• Omnitruncated 7-simplex honeycomb
• E7 honeycomb

Notes

1. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A7.html
2. N.W. Johnson: Geometries and Transformations, (2015) Chapter 12: Euclidean symmetry groups, p 177
3. 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]