Omnitruncated 5-simplex honeycomb(No image)TypeUniform honeycombFamilyOmnitruncated simplectic honeycombSchläfli symbolt012345{3}Coxeter–Dynkin diagram5-face typest01234{3,3,3,3} 4-face typest0123{3,3,3}{}×t012{3,3}{6}×{6}Cell typest012{3,3}{4,3}{}x{6}Face types{4}{6}Vertex figureIrr. 5-simplexSymmetry×12, [6[3]]Propertiesvertex-transitive In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-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).
A5* lattice The A lattice (also called A) is the union of six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.
∪ ∪ ∪ ∪ ∪ = dual of
Related polytopes and honeycombs
Projection by folding
The omnitruncated 5-simplex honeycomb can be projected into the 3-dimensional 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, (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]