7-simplex honeycomb

7-simplex honeycomb
(No image)
TypeUniform honeycomb
FamilySimplectic honeycomb
Schläfli symbol{3[8]}
Coxeter 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 figuret0,6{36}
Symmetry{\tilde{A}}_7×2, <[3[8]]>
Propertiesvertex-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
diagram		 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

See also

Regular and uniform honeycombs in 7-space:

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