- Source: Omnitruncated 5-simplex honeycomb
In five-dimensional Euclidean geometry, the 5.180.24.3/info/omnitruncated" target="_blank">omnitruncated 5-simplex honeycomb or 5.180.24.3/info/omnitruncated" target="_blank">omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of 5.180.24.3/info/omnitruncated" target="_blank">omnitruncated 5-simplex facets.
The facets of all 5.180.24.3/info/omnitruncated" target="_blank">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).
A5* lattice
The A*5 lattice (also called A65) is the union of six A5 lattices, and is the dual vertex arrangement to the 5.180.24.3/info/omnitruncated" target="_blank">omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an 5.180.24.3/info/omnitruncated" target="_blank">omnitruncated 5-simplex.
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= dual of
Related polytopes and honeycombs
This honeycomb is one of 12 unique uniform honeycombs constructed by the
A
~
5
{\displaystyle {\tilde {A}}_{5}}
Coxeter group. The extended symmetry of the hexagonal diagram of the
A
~
5
{\displaystyle {\tilde {A}}_{5}}
Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:
= Projection by folding
=The 5.180.24.3/info/omnitruncated" target="_blank">omnitruncated 5-simplex honeycomb can be projected into the 3-dimensional 5.180.24.3/info/omnitruncated" target="_blank">omnitruncated cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same 3-space vertex arrangement:
See also
Regular and uniform honeycombs in 5-space:
5-cube honeycomb
5-demicube honeycomb
5-simplex 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]