Hexicated 7-simplexes
![]() 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 |
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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
Hexicated 7-simplex | |
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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.
![](/wiki/images/thumb/a/a3/Ammann-Beenker_tiling_example.png/250px-Ammann-Beenker_tiling_example.png)
Root vectors
Its 56 vertices represent the root vectors of the simple Lie group A7.
Alternate names
- Expanded 7-simplex
- Small petated hexadecaexon (acronym: suph) (Jonathan Bowers)[1]
Coordinates
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:
- (1,-1,0,0,0,0,0,0)
Images
Hexitruncated 7-simplex
hexitruncated 7-simplex | |
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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 |
Alternate names
- Petitruncated octaexon (acronym: puto) (Jonathan Bowers)[2]
Coordinates
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, .
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Hexicantellated 7-simplex
Hexicantellated 7-simplex | |
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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 |
Alternate names
- Petirhombated octaexon (acronym: puro) (Jonathan Bowers)[3]
Coordinates
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, .
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Hexiruncinated 7-simplex
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 |
Alternate names
- Petiprismated hexadecaexon (acronym: puph) (Jonathan Bowers)[4]
Coordinates
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, .
Images
Hexicantitruncated 7-simplex
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 |
Alternate names
- Petigreatorhombated octaexon (acronym: pugro) (Jonathan Bowers)[5]
Coordinates
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, .
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Hexiruncitruncated 7-simplex
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 |
Alternate names
- Petiprismatotruncated octaexon (acronym: pupato) (Jonathan Bowers)[6]
Coordinates
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, .
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Hexiruncicantellated 7-simplex
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.
Alternate names
- Petiprismatorhombated octaexon (acronym: pupro) (Jonathan Bowers)[7]
Coordinates
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, .
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Hexisteritruncated 7-simplex
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 |
Alternate names
- Peticellitruncated octaexon (acronym: pucto) (Jonathan Bowers)[8]
Coordinates
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, .
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Hexistericantellated 7-simplex
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 |
Alternate names
- Peticellirhombihexadecaexon (acronym: pucroh) (Jonathan Bowers)[9]
Coordinates
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, .
Images
Hexipentitruncated 7-simplex
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 |
Alternate names
- Petiteritruncated hexadecaexon (acronym: putath) (Jonathan Bowers)[10]
Coordinates
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, .
Images
Hexiruncicantitruncated 7-simplex
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 |
Alternate names
- Petigreatoprismated octaexon (acronym: pugopo) (Jonathan Bowers)[11]
Coordinates
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, .
Images
Hexistericantitruncated 7-simplex
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 |
Alternate names
- Peticelligreatorhombated octaexon (acronym: pucagro) (Jonathan Bowers)[12]
Coordinates
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, .
Images
Hexisteriruncitruncated 7-simplex
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 |
Alternate names
- Peticelliprismatotruncated octaexon (acronym: pucpato) (Jonathan Bowers)[13]
Coordinates
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, .
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Hexisteriruncicantellated 7-simplex
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 |
Alternate names
- Peticelliprismatorhombihexadecaexon (acronym: pucproh) (Jonathan Bowers)[14]
Coordinates
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, .
Images
Hexipenticantitruncated 7-simplex
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 |
Alternate names
- Petiterigreatorhombated octaexon (acronym: putagro) (Jonathan Bowers)[15]
Coordinates
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, .
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Hexipentiruncitruncated 7-simplex
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 |
Alternate names
- Petiteriprismatotruncated hexadecaexon (acronym: putpath) (Jonathan Bowers)[16]
Coordinates
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, .
Images
Hexisteriruncicantitruncated 7-simplex
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 |
Alternate names
- Petigreatocellated octaexon (acronym: pugaco) (Jonathan Bowers)[17]
Coordinates
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, .
Images
Hexipentiruncicantitruncated 7-simplex
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 |
Alternate names
- Petiterigreatoprismated octaexon (acronym: putgapo) (Jonathan Bowers)[18]
Coordinates
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, .
Images
Hexipentistericantitruncated 7-simplex
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 |
Alternate names
- Petitericelligreatorhombihexadecaexon (acronym: putcagroh) (Jonathan Bowers)[19]
Coordinates
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, .
Images
Omnitruncated 7-simplex
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 .
Alternate names
- Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers)[20]
Coordinates
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}, .
Images
Related polytopes
These polytope are a part of 71 uniform 7-polytopes with A7 symmetry.
Notes
- ↑ Klitzing, (x3o3o3o3o3o3x - suph)
- ↑ Klitzing, (x3x3o3o3o3o3x- puto)
- ↑ Klitzing, (x3o3x3o3o3o3x - puro)
- ↑ Klitzing, (x3o3o3x3o3o3x - puph)
- ↑ Klitzing, (x3o3o3o3x3o3x - pugro)
- ↑ Klitzing, (x3x3x3o3o3o3x - pupato)
- ↑ Klitzing, (x3o3x3x3o3o3x - pupro)
- ↑ Klitzing, (x3x3o3o3x3o3x - pucto)
- ↑ Klitzing, (x3o3x3o3x3o3x - pucroh)
- ↑ Klitzing, (x3x3o3o3o3x3x - putath)
- ↑ Klitzing, (x3x3x3x3o3o3x - pugopo)
- ↑ Klitzing, (x3x3x3o3x3o3x - pucagro)
- ↑ Klitzing, (x3x3o3x3x3o3x - pucpato)
- ↑ Klitzing, (x3o3x3x3x3o3x - pucproh)
- ↑ Klitzing, (x3x3x3o3o3x3x - putagro)
- ↑ Klitzing, (x3x3o3x3o3x3x - putpath)
- ↑ Klitzing, (x3x3x3x3x3o3x - pugaco)
- ↑ Klitzing, (x3x3x3x3o3x3x - putgapo)
- ↑ Klitzing, (x3x3x3o3x3x3x - putcagroh)
- ↑ Klitzing, (x3x3x3x3x3x3x - guph)
References
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN:978-0-471-01003-6, wiley.com
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, PhD (1966)
- Klitzing, Richard. "7D". https://bendwavy.org/klitzing/dimensions/polyexa.htm. x3o3o3o3o3o3x - suph, x3x3o3o3o3o3x- puto, x3o3x3o3o3o3x - puro, x3o3o3x3o3o3x - puph, x3o3o3o3x3o3x - pugro, x3x3x3o3o3o3x - pupato, x3o3x3x3o3o3x - pupro, x3x3o3o3x3o3x - pucto, x3o3x3o3x3o3x - pucroh, x3x3o3o3o3x3x - putath, x3x3x3x3o3o3x - pugopo, x3x3x3o3x3o3x - pucagro, x3x3o3x3x3o3x - pucpato, x3o3x3x3x3o3x - pucproh, x3x3x3o3o3x3x - putagro, x3x3x3x3o3x3x - putpath, x3x3x3x3x3o3x - pugaco, x3x3x3x3o3x3x - putgapo, x3x3x3o3x3x3x - putcagroh, x3x3x3x3x3x3x - guph
External links
Fundamental convex regular and uniform polytopes in dimensions 2–10
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Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |
![]() | Original source: https://en.wikipedia.org/wiki/Hexicated 7-simplexes.
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