Omnitruncated simplicial honeycomb


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.
nImageTessellationFacetsVertex figureFacets per vertex figureVertices per vertex figure
1Apeirogon
Line segmentLine segment12
2Hexagonal tiling

hexagon		Equilateral triangle
3 hexagons3
3Bitruncated cubic honeycomb

Truncated octahedron		irr. tetrahedron
4 truncated octahedron4
4Omnitruncated 4-simplex honeycomb

Omnitruncated 4-simplex		irr. 5-cell
5 omnitruncated 4-simplex5
5Omnitruncated 5-simplex honeycomb

Omnitruncated 5-simplex		irr. 5-simplex
6 omnitruncated 5-simplex6
6Omnitruncated 6-simplex honeycomb

Omnitruncated 6-simplex		irr. 6-simplex
7 omnitruncated 6-simplex7
7Omnitruncated 7-simplex honeycomb

Omnitruncated 7-simplex		irr. 7-simplex
8 omnitruncated 7-simplex8
8Omnitruncated 8-simplex honeycomb

Omnitruncated 8-simplex		irr. 8-simplex
9 omnitruncated 8-simplex9

Projection by folding

The -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:
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