In geometry an omnitruncated simplicial honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry of the affine Coxeter group. Each is composed of omnitruncated simplex facets. The vertex figure for each is an irregular n-simplex.
The facets of an omnitruncated simplicial honeycomb are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,...,n).
nImageTessellationFacetsVertex figureFacets per vertex figureVertices per vertex figure1ApeirogonLine segmentLine segment122Hexagonal tilinghexagonEquilateral triangle3 hexagons33Bitruncated cubic honeycombTruncated octahedronirr. tetrahedron4 truncated octahedron44Omnitruncated 4-simplex honeycombOmnitruncated 4-simplexirr. 5-cell5 omnitruncated 4-simplex55Omnitruncated 5-simplex honeycombOmnitruncated 5-simplexirr. 5-simplex6 omnitruncated 5-simplex66Omnitruncated 6-simplex honeycombOmnitruncated 6-simplexirr. 6-simplex7 omnitruncated 6-simplex77Omnitruncated 7-simplex honeycombOmnitruncated 7-simplexirr. 7-simplex8 omnitruncated 7-simplex88Omnitruncated 8-simplex honeycombOmnitruncated 8-simplexirr. 8-simplex9 omnitruncated 8-simplex9
Projection by folding
The (2n−1)-simplex honeycombs can be projected into the n-dimensional omnitruncated hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
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See also Hypercubic honeycomb Alternated hypercubic honeycomb Quarter hypercubic honeycomb Simplectic honeycomb Truncated simplicial honeycomb
References George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56. Norman Johnson Uniform Polytopes, Manuscript (1991) Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, 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]