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 A_{7} Coxeter plane |
---|
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.
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 | t_{0,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | 254: 8+8 {3^{5}} 28+28 {}x{3^{4}} 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 | A_{7}×2, [[3^{6}]], 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 A_{7}.
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:
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
hexitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 1848 |
Vertices | 336 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Hexicantellated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,2,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 5880 |
Vertices | 840 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Hexiruncinated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,3,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 8400 |
Vertices | 1120 |
Vertex figure | |
Coxeter group | A_{7}, [[3^{6}]], order 80640 |
Properties | convex |
The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Hexicantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,2,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 8400 |
Vertices | 1680 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Hexiruncitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,3,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 20160 |
Vertices | 3360 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Hexiruncicantellated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,2,3,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 16800 |
Vertices | 3360 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
hexisteritruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,4,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 20160 |
Vertices | 3360 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
hexistericantellated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,2,4,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | t_{0,2,4}{3,3,3,3,3} {}xt_{0,2,4}{3,3,3,3} |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 30240 |
Vertices | 5040 |
Vertex figure | |
Coxeter group | A_{7}, [[3^{6}]], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Hexipentitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,5,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 8400 |
Vertices | 1680 |
Vertex figure | |
Coxeter group | A_{7}, [[3^{6}]], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Hexiruncicantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,2,3,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 30240 |
Vertices | 6720 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Hexistericantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,2,4,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 50400 |
Vertices | 10080 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Hexisteriruncitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,3,4,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 45360 |
Vertices | 10080 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Hexisteriruncitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,2,3,4,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 45360 |
Vertices | 10080 |
Vertex figure | |
Coxeter group | A_{7}, [[3^{6}]], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
hexipenticantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,2,5,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 30240 |
Vertices | 6720 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Hexisteriruncicantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,2,3,5,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 80640 |
Vertices | 20160 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Hexisteriruncicantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,2,3,4,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 80640 |
Vertices | 20160 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Hexipentiruncicantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,2,3,5,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 80640 |
Vertices | 20160 |
Vertex figure | |
Coxeter group | A_{7}, [3^{6}], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Hexipentistericantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,2,4,5,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 80640 |
Vertices | 20160 |
Vertex figure | |
Coxeter group | A_{7}, [[3^{6}]], 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, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Omnitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t_{0,1,2,3,4,5,6}{3^{6}} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 141120 |
Vertices | 40320 |
Vertex figure | Irr. 6-simplex |
Coxeter group | A_{7}, [[3^{6}]], order 80640 |
Properties | convex |
The omnitruncated 7-simplex is composed of 40320 (8 factorial) vertices and is the largest uniform 7-polytope in the A_{7} 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, t_{0,1,2,3,4,5,6}{3^{6},4}, .
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
These polytope are a part of 71 uniform 7-polytopes with A_{7} symmetry.