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7-simplex honeycomb
(No image)
Type	Uniform 7-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3} = 0
Coxeter diagram
6-face types	{3} , t1{3} t2{3} , t3{3}
6-face types	{3} , t1{3} t2{3}
5-face types	{3} , t1{3} t2{3}
4-face types	{3} , t1{3}
Cell types	{3,3} , t1{3,3}
Face types	{3}
Vertex figure	t0,6{3}
Symmetry	${\displaystyle {\tilde {A}}_{7}}$×21, <>
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 ${\displaystyle {\tilde {A}}_{7}}$ Coxeter group. 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.

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

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

∪ = .

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

∪ ∪ ∪ = + = dual of .

The A
₇ lattice (also called A
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 constructed by the ${\displaystyle {\tilde {A}}_{7}}$ 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			${\displaystyle {\tilde {A}}_{7}}$
d2	<>		${\displaystyle {\tilde {A}}_{7}}$×21	1
p2	]		${\displaystyle {\tilde {A}}_{7}}$×22	2
d4	<2>		${\displaystyle {\tilde {A}}_{7}}$×41
p4	]		${\displaystyle {\tilde {A}}_{7}}$×42
d8	]		${\displaystyle {\tilde {A}}_{7}}$×8
r16	]		${\displaystyle {\tilde {A}}_{7}}$×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:

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

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. "The Lattice A7".
2. N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
3. Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence 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, (1.9 Uniform space-fillings)
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E	Uniform tiling	0	δ3	hδ3	qδ3	Hexagonal
E	Uniform convex honeycomb	0	δ4	hδ4	qδ4
E	Uniform 4-honeycomb	0	δ5	hδ5	qδ5	24-cell honeycomb
E	Uniform 5-honeycomb	0	δ6	hδ6	qδ6
E	Uniform 6-honeycomb	0	δ7	hδ7	qδ7	222
E	Uniform 7-honeycomb	0	δ8	hδ8	qδ8	133 • 331
E	Uniform 8-honeycomb	0	δ9	hδ9	qδ9	152 • 251 • 521
E	Uniform 9-honeycomb	0	δ10	hδ10	qδ10
E	Uniform 10-honeycomb	0	δ11	hδ11	qδ11
E	Uniform (n-1)-honeycomb	0	δn	hδn	qδn	1k2 • 2k1 • k21

• Honeycombs (geometry)
• 8-polytopes