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).
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 |
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:
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
... | ||||
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![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
... |
Space | Family | / / | ||||
---|---|---|---|---|---|---|
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 | 133 • 331 |
E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)- honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |
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).
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 |
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:
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
... | ||||
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
... |
Space | Family | / / | ||||
---|---|---|---|---|---|---|
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 | 133 • 331 |
E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)- honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |