Pentellated 7-simplexes


7-simplex

Pentellated 7-simplex

Pentitruncated 7-simplex

Penticantellated 7-simplex

Penticantitruncated 7-simplex

Pentiruncinated 7-simplex

Pentiruncitruncated 7-simplex

Pentiruncicantellated 7-simplex

Pentiruncicantitruncated 7-simplex

Pentistericated 7-simplex

Pentisteritruncated 7-simplex

Pentistericantellated 7-simplex

Pentistericantitruncated 7-simplex

Pentisteriruncinated 7-simplex

Pentisteriruncitruncated 7-simplex

Pentisteriruncicantellated 7-simplex

Pentisteriruncicantitruncated 7-simplex

In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-simplex.

There are 16 unique pentellations of the 7-simplex with permutations of truncations, cantellations, runcinations, and sterications.

Pentellated 7-simplex

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Pentellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 1260
Vertices 168
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Small terated octaexon (acronym: seto) (Jonathan Bowers)[1]

Coordinates

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The vertices of the pentellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,1,2). This construction is based on facets of the pentellated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentitruncated 7-simplex

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pentitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 5460
Vertices 840
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Teritruncated octaexon (acronym: teto) (Jonathan Bowers)[2]

Coordinates

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The vertices of the pentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,3). This construction is based on facets of the pentitruncated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Penticantellated 7-simplex

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Penticantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 11760
Vertices 1680
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Terirhombated octaexon (acronym: tero) (Jonathan Bowers)[3]

Coordinates

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The vertices of the penticantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,2,3). This construction is based on facets of the penticantellated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Penticantitruncated 7-simplex

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penticantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Terigreatorhombated octaexon (acronym: tegro) (Jonathan Bowers)[4]

Coordinates

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The vertices of the penticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,3,4). This construction is based on facets of the penticantitruncated 8-orthoplex.

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentiruncinated 7-simplex

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pentiruncinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 10920
Vertices 1680
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Teriprismated octaexon (acronym: tepo) (Jonathan Bowers)[5]

Coordinates

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The vertices of the pentiruncinated 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 pentiruncinated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentiruncitruncated 7-simplex

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pentiruncitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 27720
Vertices 5040
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Teriprismatotruncated octaexon (acronym: tapto) (Jonathan Bowers)[6]

Coordinates

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The vertices of the pentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,3,4). This construction is based on facets of the pentiruncitruncated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentiruncicantellated 7-simplex

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pentiruncicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 25200
Vertices 5040
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Teriprismatorhombated octaexon (acronym: tapro) (Jonathan Bowers)[7]

Coordinates

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The vertices of the pentiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,3,4). This construction is based on facets of the pentiruncicantellated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentiruncicantitruncated 7-simplex

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pentiruncicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 45360
Vertices 10080
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Terigreatoprismated octaexon (acronym: tegapo) (Jonathan Bowers)[8]

Coordinates

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The vertices of the pentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,4,5). This construction is based on facets of the pentiruncicantitruncated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentistericated 7-simplex

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pentistericated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 4200
Vertices 840
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Tericellated octaexon (acronym: teco) (Jonathan Bowers)[9]

Coordinates

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The vertices of the pentistericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,2,3). This construction is based on facets of the pentistericated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentisteritruncated 7-simplex

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pentisteritruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 15120
Vertices 3360
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Tericellitruncated octaexon (acronym: tecto) (Jonathan Bowers)[10]

Coordinates

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The vertices of the pentisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,4). This construction is based on facets of the pentisteritruncated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentistericantellated 7-simplex

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pentistericantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 25200
Vertices 5040
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Tericellirhombated octaexon (acronym: tecro) (Jonathan Bowers)[11]

Coordinates

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The vertices of the pentistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,3,4). This construction is based on facets of the pentistericantellated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentistericantitruncated 7-simplex

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pentistericantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 40320
Vertices 10080
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Tericelligreatorhombated octaexon (acronym: tecagro) (Jonathan Bowers)[12]

Coordinates

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The vertices of the pentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,5). This construction is based on facets of the pentistericantitruncated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentisteriruncinated 7-simplex

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Pentisteriruncinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,3,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 15120
Vertices 3360
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Bipenticantitruncated 7-simplex as t1,2,3,6{3,3,3,3,3,3}
  • Tericelliprismated octaexon (acronym: tacpo) (Jonathan Bowers)[13]

Coordinates

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The vertices of the pentisteriruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,3,3,4). This construction is based on facets of the pentisteriruncinated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Pentisteriruncitruncated 7-simplex

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pentisteriruncitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,3,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 40320
Vertices 10080
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Tericelliprismatotruncated octaexon (acronym: tacpeto) (Jonathan Bowers)[14]

Coordinates

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The vertices of the pentisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,3,4,5). This construction is based on facets of the pentisteriruncitruncated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [[5]] [4] [[3]]

Pentisteriruncicantellated 7-simplex

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pentisteriruncicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,3,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 40320
Vertices 10080
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Bipentiruncicantitruncated 7-simplex as t1,2,3,4,6{3,3,3,3,3,3}
  • Tericelliprismatorhombated octaexon (acronym: tacpro) (Jonathan Bowers)[15]

Coordinates

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The vertices of the pentisteriruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,4,5). This construction is based on facets of the pentisteriruncicantellated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [[5]] [4] [[3]]

Pentisteriruncicantitruncated 7-simplex

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pentisteriruncicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 70560
Vertices 20160
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate names

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  • Great terated octaexon (acronym: geto) (Jonathan Bowers)[16]

Coordinates

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The vertices of the pentisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,5,6). This construction is based on facets of the pentisteriruncicantitruncated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [[5]] [4] [[3]]
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These polytopes are a part of a set of 71 uniform 7-polytopes with A7 symmetry.

A7 polytopes
 
t0
 
t1
 
t2
 
t3
 
t0,1
 
t0,2
 
t1,2
 
t0,3
 
t1,3
 
t2,3
 
t0,4
 
t1,4
 
t2,4
 
t0,5
 
t1,5
 
t0,6
 
t0,1,2
 
t0,1,3
 
t0,2,3
 
t1,2,3
 
t0,1,4
 
t0,2,4
 
t1,2,4
 
t0,3,4
 
t1,3,4
 
t2,3,4
 
t0,1,5
 
t0,2,5
 
t1,2,5
 
t0,3,5
 
t1,3,5
 
t0,4,5
 
t0,1,6
 
t0,2,6
 
t0,3,6
 
t0,1,2,3
 
t0,1,2,4
 
t0,1,3,4
 
t0,2,3,4
 
t1,2,3,4
 
t0,1,2,5
 
t0,1,3,5
 
t0,2,3,5
 
t1,2,3,5
 
t0,1,4,5
 
t0,2,4,5
 
t1,2,4,5
 
t0,3,4,5
 
t0,1,2,6
 
t0,1,3,6
 
t0,2,3,6
 
t0,1,4,6
 
t0,2,4,6
 
t0,1,5,6
 
t0,1,2,3,4
 
t0,1,2,3,5
 
t0,1,2,4,5
 
t0,1,3,4,5
 
t0,2,3,4,5
 
t1,2,3,4,5
 
t0,1,2,3,6
 
t0,1,2,4,6
 
t0,1,3,4,6
 
t0,2,3,4,6
 
t0,1,2,5,6
 
t0,1,3,5,6
 
t0,1,2,3,4,5
 
t0,1,2,3,4,6
 
t0,1,2,3,5,6
 
t0,1,2,4,5,6
 
t0,1,2,3,4,5,6

Notes

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  1. ^ Klitzing, (x3o3o3o3o3x3o - seto)
  2. ^ Klitzing, (x3x3o3o3o3x3o - teto)
  3. ^ Klitzing, (x3o3x3o3o3x3o - tero)
  4. ^ Klitzing, (x3x3x3oxo3x3o - tegro)
  5. ^ Klitzing, (x3o3o3x3o3x3o - tepo)
  6. ^ Klitzing, (x3x3o3x3o3x3o - tapto)
  7. ^ Klitzing, (x3o3x3x3o3x3o - tapro)
  8. ^ Klitzing, (x3x3x3x3o3x3o - tegapo)
  9. ^ Klitzing, (x3o3o3o3x3x3o - teco)
  10. ^ Klitzing, (x3x3o3o3x3x3o - tecto)
  11. ^ Klitzing, (x3o3x3o3x3x3o - tecro)
  12. ^ Klitzing, (x3x3x3o3x3x3o - tecagro)
  13. ^ Klitzing, (x3o3o3x3x3x3o - tacpo)
  14. ^ Klitzing, (x3x3o3x3x3x3o - tacpeto)
  15. ^ Klitzing, (x3o3x3x3x3x3o - tacpro)
  16. ^ Klitzing, (x3x3x3x3x3x3o - geto)

References

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  • 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 [1]
      • (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, Ph.D.
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3o3o3x3o - seto, x3x3o3o3o3x3o - teto, x3o3x3o3o3x3o - tero, x3x3x3oxo3x3o - tegro, x3o3o3x3o3x3o - tepo, x3x3o3x3o3x3o - tapto, x3o3x3x3o3x3o - tapro, x3x3x3x3o3x3o - tegapo, x3o3o3o3x3x3o - teco, x3x3o3o3x3x3o - tecto, x3o3x3o3x3x3o - tecro, x3x3x3o3x3x3o - tecagro, x3o3o3x3x3x3o - tacpo, x3x3o3x3x3x3o - tacpeto, x3o3x3x3x3x3o - tacpro, x3x3x3x3x3x3o - geto
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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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds