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).
Projection by folding
The (2nâÂÂ1)-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:
See also
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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
- (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]