7-simplex honeycomb

7-simplex honeycomb
(No image)
Type	Uniform 7-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3[8]} = 0[8]
Coxeter diagram
6-face types	t3{36}
6-face types	t2{35}
5-face types	t2{34}
4-face types	{33} , t1{33}
Cell types	{3,3} , t1{3,3}
Face types	{3}
Vertex figure	t0,6{36}
Symmetry	${\displaystyle {\tilde {A}}_{7}}$×21, <[3[8]]>
Properties	vertex-transitive

In

facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.

This

represent the 56 roots of the ${\displaystyle {\tilde {A}}_{7}}$ Coxeter group..

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

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

.

∪ = .

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

.

∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .

This honeycomb is one of 29 unique uniform honeycombs^[3] constructed by the ${\displaystyle {\tilde {A}}_{7}}$

diagram:

A7 honeycombs
Octagon symmetry	Extended symmetry	Extended diagram	Extended group	Honeycombs
a1	[3[8]]		${\displaystyle {\tilde {A}}_{7}}$
d2	<[3[8]]>		${\displaystyle {\tilde {A}}_{7}}$×21	1
p2	[[3[8]]]		${\displaystyle {\tilde {A}}_{7}}$×22	2
d4	<2[3[8]]>		${\displaystyle {\tilde {A}}_{7}}$×41
p4	[2[3[8]]]		${\displaystyle {\tilde {A}}_{7}}$×42
d8	[4[3[8]]]		${\displaystyle {\tilde {A}}_{7}}$×8
r16	[8[3[8]]]		${\displaystyle {\tilde {A}}_{7}}$×16	3

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:

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

Regular and uniform honeycombs in 7-space:

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

• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Kaleidoscopes: Selected Writings of
  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]

regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
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