- Source: Omnitruncated 7-simplex honeycomb
In seven-dimensional Euclidean geometry, the omnitruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 7-simplex facets.
The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
A7* lattice
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.
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= 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:
See also
Regular and uniform honeycombs in 7-space:
7-cubic honeycomb
7-demicubic honeycomb
7-simplex honeycomb
Truncated 7-simplex honeycomb
331 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]