Runcinated 7-simplexes

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7-simplex

Runcinated 7-simplex

Biruncinated 7-simplex

Runcitruncated 7-simplex

Biruncitruncated 7-simplex

Runcicantellated 7-simplex

Biruncicantellated 7-simplex

Runcicantitruncated 7-simplex

Biruncicantitruncated 7-simplex
Orthogonal projections in A7 Coxeter plane

In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.

There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations.

Runcinated 7-simplex

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Runcinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 2100
Vertices 280
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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  • Small prismated octaexon (acronym: spo) (Jonathan Bowers)[1]

Coordinates

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

Biruncinated 7-simplex

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Biruncinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 4200
Vertices 560
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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  • Small biprismated octaexon (sibpo) (Jonathan Bowers)[2]

Coordinates

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

Runcitruncated 7-simplex

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runcitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 4620
Vertices 840
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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  • Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)[3]

Coordinates

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

Biruncitruncated 7-simplex

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Biruncitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,4{3,3,3,3,3,3}
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

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  • Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)[4]

Coordinates

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

Runcicantellated 7-simplex

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runcicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 840
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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  • Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)[5]

Coordinates

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

Biruncicantellated 7-simplex

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biruncicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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  • Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)

Coordinates

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

Runcicantitruncated 7-simplex

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runcicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 5880
Vertices 1680
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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  • Great prismated octaexon (acronym: gapo) (Jonathan Bowers)[6]

Coordinates

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

Biruncicantitruncated 7-simplex

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biruncicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 11760
Vertices 3360
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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  • Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)[7]

Coordinates

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The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 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 among 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, (x3o3o3x3o3o3o - spo)
  2. ^ Klitzing, (o3x3o3o3x3o3o - sibpo)
  3. ^ Klitzing, (x3x3o3x3o3o3o - patto)
  4. ^ Klitzing, (o3x3x3o3x3o3o - bipto)
  5. ^ Klitzing, (x3o3x3x3o3o3o - paro)
  6. ^ Klitzing, (x3x3x3x3o3o3o - gapo)
  7. ^ Klitzing, (o3x3x3x3x3o3o- gibpo)

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)". x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o- gibpo
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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