Omnitruncated simplicial honeycomb explained

In geometry an omnitruncated simplicial honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry of the

{\tilde{A}} _{n} 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).

height=30	n	{\tilde{A}} _{1+}	Image	Tessellation	Facets	Vertex figure	Facets per vertex figure	Vertices per vertex figure
1	{\tilde{A}} _{1}		Apeirogon	Line segment	Line segment	1	2
2	{\tilde{A}} _{2}		Hexagonal tiling	hexagon	Equilateral triangle	3 hexagons	3
3	{\tilde{A}} _{3}		Bitruncated cubic honeycomb	Truncated octahedron	irr. tetrahedron	4 truncated octahedron	4
4	{\tilde{A}} _{4}		Omnitruncated 4-simplex honeycomb	Omnitruncated 4-simplex	irr. 5-cell	5 omnitruncated 4-simplex	5
5	{\tilde{A}} _{5}		Omnitruncated 5-simplex honeycomb	Omnitruncated 5-simplex	irr. 5-simplex	6 omnitruncated 5-simplex	6
6	{\tilde{A}} _{6}		Omnitruncated 6-simplex honeycomb	Omnitruncated 6-simplex	irr. 6-simplex	7 omnitruncated 6-simplex	7
7	{\tilde{A}} _{7}		Omnitruncated 7-simplex honeycomb	Omnitruncated 7-simplex	irr. 7-simplex	8 omnitruncated 7-simplex	8
8	{\tilde{A}} _{8}		Omnitruncated 8-simplex honeycomb	Omnitruncated 8-simplex	irr. 8-simplex	9 omnitruncated 8-simplex	9	-	9	{\tilde{A}} _{9}		Omnitruncated 9-simplex honeycomb	Omnitruncated 9-simplex	irr. 9-simplex	10 omnitruncated 9-simplex	10
10	{\tilde{A}} _{10}		Omnitruncated 10-simplex honeycomb	Omnitruncated 10-simplex	irr. 10-simplex	11 omnitruncated 10-simplex	11-->

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:

{\tilde{A}} _{3}		{\tilde{A}} _{5}		{\tilde{A}} _{7}		{\tilde{A}} _{9}		...
{\tilde{C}} _{2}		{\tilde{C}} _{3}		{\tilde{C}} _{4}		{\tilde{C}} _{5}		...

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, 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]