7-simplex honeycomb explained

bgcolor=#e7dcc3 colspan=2	7-simplex honeycomb
bgcolor=#ffffff align=center colspan=2	(No image)
Type	Uniform 7-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	= 0[8]
Coxeter diagram
6-face types
6-face types
5-face types
4-face types
Cell types
Face types
Vertex figure
Symmetry	{\tilde{A}} _{7} ×21, <[3<sup>[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 A lattice can be constructed as the union of two A₇ lattices, and is identical to the E7 lattice.

∪ = .

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

∪ ∪ ∪ = + = dual of .

The A lattice (also called A) 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

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:

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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  • (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]

Notes and References

1. Web site: The Lattice A7.
2. N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294