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Type of 7-polytope From Wikipedia, the free encyclopedia
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.
7-simplex    |
 Hexicated 7-simplex |
 Hexitruncated 7-simplex |
 Hexicantellated 7-simplex |
 Hexiruncinated 7-simplex |
 Hexicantitruncated 7-simplex |
 Hexiruncitruncated 7-simplex |
 Hexiruncicantellated 7-simplex |
 Hexisteritruncated 7-simplex |
 Hexistericantellated 7-simplex |
 Hexipentitruncated 7-simplex |
 Hexiruncicantitruncated 7-simplex |
 Hexistericantitruncated 7-simplex |
 Hexisteriruncitruncated 7-simplex |
 Hexisteriruncicantellated 7-simplex |
 Hexipenticantitruncated 7-simplex |
 Hexipentiruncitruncated 7-simplex |
 Hexisteriruncicantitruncated 7-simplex |
 Hexipentiruncicantitruncated 7-simplex |
 Hexipentistericantitruncated 7-simplex |
 Hexipentisteriruncicantitruncated 7-simplex (Omnitruncated 7-simplex) | |||
| Orthogonal projections in A7 Coxeter plane | |||
|---|---|---|---|
There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations.
The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex. The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called a omnitruncated 7-simplex with all of the nodes ringed.
| Hexicated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 254: 8+8 {35}  28+28 {}x{34} 56+56 {3}x{3,3,3} 70 {3,3}x{3,3} |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 336 |
| Vertices | 56 |
| Vertex figure | 5-simplex antiprism |
| Coxeter group | A7×2, [[36]], order 80640 |
| Properties | convex |
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an expansion operation.
Its 56 vertices represent the root vectors of the simple Lie group A7.
The vertices of the hexicated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,2). This construction is based on facets of the hexicated 8-orthoplex, 
.
A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations of:
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| hexitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1848 |
| Vertices | 336 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [5] | [4] | [3] |
| Hexicantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5880 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [5] | [4] | [3] |
| Hexiruncinated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,3,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8400 |
| Vertices | 1120 |
| Vertex figure | |
| Coxeter group | A7×2, [[36]], order 80640 |
| Properties | convex |
The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| Hexicantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8400 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [5] | [4] | [3] |
| Hexiruncitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,3,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 20160 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on facets of the hexiruncitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [5] | [4] | [3] |
| Hexiruncicantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,3,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 16800 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
In seven-dimensional geometry, a hexiruncicantellated 7-simplex is a uniform 7-polytope.
The vertices of the hexiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,3,3,4). This construction is based on facets of the hexiruncicantellated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [5] | [4] | [3] |
| hexisteritruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,4,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 20160 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,2,3,4). This construction is based on facets of the hexisteritruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [5] | [4] | [3] |
| hexistericantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,4,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | t0,2,4{3,3,3,3,3} {}xt0,2,4{3,3,3,3} |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30240 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter group | A7×2, [[36]], order 80640 |
| Properties | convex |
The vertices of the hexistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,3,4). This construction is based on facets of the hexistericantellated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| Hexipentitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,5,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8400 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A7×2, [[36]], order 80640 |
| Properties | convex |
The vertices of the hexipentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,2,3,4). This construction is based on facets of the hexipentitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| Hexiruncicantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,3,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30240 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexiruncicantitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| Hexistericantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,4,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 50400 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexistericantitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| Hexisteriruncitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,3,4,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 45360 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,3,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [5] | [4] | [3] |
| Hexisteriruncicantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,3,4,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 45360 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter group | A7×2, [[36]], order 80640 |
| Properties | convex |
The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| hexipenticantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,5,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30240 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexipenticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,3,4,5). This construction is based on facets of the hexipenticantitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [5] | [4] | [3] |
| Hexipentiruncitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,3,5,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter group | A7×2, [[36]], order 80640 |
| Properties | convex |
The vertices of the hexipentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,4,5). This construction is based on facets of the hexipentiruncitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| Hexisteriruncicantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,3,4,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 80640 |
| Vertices | 20160 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,5,6). This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| Hexipentiruncicantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,3,5,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 80640 |
| Vertices | 20160 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
The vertices of the hexipentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,5,6). This construction is based on facets of the hexipentiruncicantitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| Hexipentistericantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,4,5,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 80640 |
| Vertices | 20160 |
| Vertex figure | |
| Coxeter group | A7×2, [[36]], order 80640 |
| Properties | convex |
The vertices of the hexipentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,3,4,5,6). This construction is based on facets of the hexipentistericantitruncated 8-orthoplex, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| Omnitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,3,4,5,6{36} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 254 |
| 5-faces | 5796 |
| 4-faces | 40824 |
| Cells | 126000 |
| Faces | 191520 |
| Edges | 141120 |
| Vertices | 40320 |
| Vertex figure | Irr. 6-simplex |
| Coxeter group | A7×2, [[36]], order 80640 |
| Properties | convex |
The omnitruncated 7-simplex is composed of 40320 (8 factorial) vertices and is the largest uniform 7-polytope in the A7 symmetry of the regular 7-simplex. It can also be called the hexipentisteriruncicantitruncated 7-simplex which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.
The omnitruncated 7-simplex is the permutohedron of order 8. The omnitruncated 7-simplex is a zonotope, the Minkowski sum of eight line segments parallel to the eight lines through the origin and the eight vertices of the 7-simplex.
Like all uniform omnitruncated n-simplices, the omnitruncated 7-simplex can tessellate space by itself, in this case 7-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of 
.
The vertices of the omnitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,4,5,6,7). This construction is based on facets of the hexipentisteriruncicantitruncated 8-orthoplex, t0,1,2,3,4,5,6{36,4}, .
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [[5]] | [4] | [[3]] |
These polytope are a part of 71 uniform 7-polytopes with A7 symmetry.
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