# Omnitruncated simplectic honeycomb

In geometry an omnitruncated simplectic honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry of the ${\displaystyle {\tilde {A}}_{n}}$ 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 ${\displaystyle {\tilde {A}}_{1+}}$ Image Tessellation Facets Vertex figure Facets per vertex figure Vertices per vertex figure
1 ${\displaystyle {\tilde {A}}_{1}}$ Apeirogon
Line segment Line segment 1 2
2 ${\displaystyle {\tilde {A}}_{2}}$ Hexagonal tiling

hexagon
Equilateral triangle
3 hexagons 3
3 ${\displaystyle {\tilde {A}}_{3}}$ Bitruncated cubic honeycomb

Truncated octahedron
irr. tetrahedron
4 truncated octahedron 4
4 ${\displaystyle {\tilde {A}}_{4}}$ Omnitruncated 4-simplex honeycomb

Omnitruncated 4-simplex
irr. 5-cell
5 omnitruncated 4-simplex 5
5 ${\displaystyle {\tilde {A}}_{5}}$ Omnitruncated 5-simplex honeycomb

Omnitruncated 5-simplex
irr. 5-simplex
6 omnitruncated 5-simplex 6
6 ${\displaystyle {\tilde {A}}_{6}}$ Omnitruncated 6-simplex honeycomb

Omnitruncated 6-simplex
irr. 6-simplex
7 omnitruncated 6-simplex 7
7 ${\displaystyle {\tilde {A}}_{7}}$ Omnitruncated 7-simplex honeycomb

Omnitruncated 7-simplex
irr. 7-simplex
8 omnitruncated 7-simplex 8
8 ${\displaystyle {\tilde {A}}_{8}}$ 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:

${\displaystyle {\tilde {A}}_{3}}$ ... ...

## 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
• (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]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$ ${\displaystyle {\tilde {C}}_{n-1}}$ ${\displaystyle {\tilde {B}}_{n-1}}$ ${\displaystyle {\tilde {D}}_{n-1}}$ ${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21