Omnitruncated simplicial honeycomb
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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 (0,1,..,n).
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n | Image | Tessellation | Facets | Vertex figure | Facets per vertex figure | Vertices per vertex figure | |
---|---|---|---|---|---|---|---|
1 | ![]() |
Apeirogon![]() ![]() ![]() |
Line segment | Line segment | 1 | 2 | |
2 | ![]() |
Hexagonal tiling![]() ![]() ![]() |
![]() hexagon |
Equilateral triangle![]() |
3 hexagons | 3 | |
3 | ![]() |
Bitruncated cubic honeycomb![]() ![]() ![]() ![]() ![]() |
![]() Truncated octahedron |
irr. tetrahedron![]() |
4 truncated octahedron | 4 | |
4 | Omnitruncated 4-simplex honeycomb![]() ![]() ![]() ![]() ![]() |
![]() Omnitruncated 4-simplex |
irr. 5-cell![]() |
5 omnitruncated 4-simplex | 5 | ||
5 | Omnitruncated 5-simplex honeycomb![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Omnitruncated 5-simplex |
irr. 5-simplex![]() |
6 omnitruncated 5-simplex | 6 | ||
6 | Omnitruncated 6-simplex honeycomb![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Omnitruncated 6-simplex |
irr. 6-simplex![]() |
7 omnitruncated 6-simplex | 7 | ||
7 | Omnitruncated 7-simplex honeycomb![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Omnitruncated 7-simplex |
irr. 7-simplex![]() |
8 omnitruncated 7-simplex | 8 | ||
8 | Omnitruncated 8-simplex honeycomb![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Omnitruncated 8-simplex |
irr. 8-simplex![]() |
9 omnitruncated 8-simplex | 9 |
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