7-simplex honeycomb

7-simplex honeycomb
(No image)
Type	Uniform 7-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3[8]} = 0[8]
Coxeter diagram	Template:CDD
6-face types	{36} File:7-simplex t0.svg, t1{36} Error creating thumbnail: t2{36} Error creating thumbnail: , t3{36} Error creating thumbnail:
6-face types	{35} , t1{35} t2{35}
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} , t1{33} File:4-simplex t1.svg
Cell types	{3,3} , t1{3,3} Error creating thumbnail:
Face types	{3}
Vertex figure	t0,6{36} Error creating thumbnail:
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. 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. Both ${\tilde{E}}_7$ and ${\tilde{A}}_7$ can be seen as affine extensions from $A_7$ from different nodes: File:Affine A7 E7 relations.png

The ATemplate:Sup sub lattice can be constructed as the union of two A₇ lattices, and is identical to the E7 lattice.

Template:CDD ∪ Template:CDD = Template:CDD.

The ATemplate:Sup sub lattice is the union of four A₇ lattices, which is identical to the E7* lattice (or ETemplate:Sup sub).

Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD = Template:CDD + Template:CDD = dual of Template:CDD.

The ATemplate:Sup sub lattice (also called ATemplate:Sup sub) 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.

Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD = dual of Template:CDD.

Related polytopes and honeycombs

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:

${\tilde{A}}_7$	Template:CDD
${\tilde{C}}_4$	Template:CDD

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

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, Template:ISBN [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]

Template:Navbar-collapsible
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