# Omnitruncated simplectic honeycomb

In geometry an omnitruncated simplectic honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellations, 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 simplectic honeycomb are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

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

## 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:

${\tilde{A}}_3$ ... ...
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## 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, ISBN 0-486-61480-8
• 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]
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