- Source: 7-simplex honeycomb
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
A
~
7
{\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.
E
~
7
{\displaystyle {\tilde {E}}_{7}}
contains
A
~
7
{\displaystyle {\tilde {A}}_{7}}
as a subgroup of index 144. Both
E
~
7
{\displaystyle {\tilde {E}}_{7}}
and
A
~
7
{\displaystyle {\tilde {A}}_{7}}
can be seen as affine extensions from
A
7
{\displaystyle A_{7}}
from different nodes:
The A27 lattice can be constructed as the union of two A7 lattices, and is identical to the E7 lattice.
∪ = .
The A47 lattice is the union of four A7 lattices, which is identical to the E7* lattice (or E27).
∪ ∪ ∪ = + = dual of .
The A*7 lattice (also called A87) 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
A
~
7
{\displaystyle {\tilde {A}}_{7}}
Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:
= 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
Notes
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 [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]