# 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 ${\tilde {A}}_{n}$ 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:

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