Runcinated 7-simplexes

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File:7-simplex t0.svg
7-simplex
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:7-simplex t03.svg
Runcinated 7-simplex
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:7-simplex t14.svg
Biruncinated 7-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:7-simplex t013.svg
Runcitruncated 7-simplex
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:7-simplex t124.svg
Biruncitruncated 7-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:7-simplex t023.svg
Runcicantellated 7-simplex
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:7-simplex t134.svg
Biruncicantellated 7-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:7-simplex t0123.svg
Runcicantitruncated 7-simplex
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:7-simplex t1234.svg
Biruncicantitruncated 7-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
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

Runcinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 2100
Vertices 280
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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

Coordinates

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

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph File:7-simplex t03.svg File:7-simplex t03 A6.svg File:7-simplex t03 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:7-simplex t03 A4.svg File:7-simplex t03 A3.svg File:7-simplex t03 A2.svg
Dihedral symmetry [5] [4] [3]

Biruncinated 7-simplex

Biruncinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 4200
Vertices 560
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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

Coordinates

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

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph File:7-simplex t14.svg File:7-simplex t14 A6.svg File:7-simplex t14 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:7-simplex t14 A4.svg File:7-simplex t14 A3.svg File:7-simplex t14 A2.svg
Dihedral symmetry [5] [4] [3]

Runcitruncated 7-simplex

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

Alternate names

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

Coordinates

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

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph File:7-simplex t013.svg File:7-simplex t013 A6.svg File:7-simplex t013 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:7-simplex t013 A4.svg File:7-simplex t013 A3.svg File:7-simplex t013 A2.svg
Dihedral symmetry [5] [4] [3]

Biruncitruncated 7-simplex

Biruncitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 8400
Vertices 1680
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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

Coordinates

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

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph File:7-simplex t124.svg File:7-simplex t124 A6.svg File:7-simplex t124 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:7-simplex t124 A4.svg File:7-simplex t124 A3.svg File:7-simplex t124 A2.svg
Dihedral symmetry [5] [4] [3]

Runcicantellated 7-simplex

runcicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 840
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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

Coordinates

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

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph File:7-simplex t023.svg File:7-simplex t023 A6.svg File:7-simplex t023 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:7-simplex t023 A4.svg File:7-simplex t023 A3.svg File:7-simplex t023 A2.svg
Dihedral symmetry [5] [4] [3]

Biruncicantellated 7-simplex

biruncicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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

Coordinates

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

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph File:7-simplex t134.svg File:7-simplex t134 A6.svg File:7-simplex t134 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:7-simplex t134 A4.svg File:7-simplex t134 A3.svg File:7-simplex t134 A2.svg
Dihedral symmetry [5] [4] [3]

Runcicantitruncated 7-simplex

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

Alternate names

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

Coordinates

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

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph File:7-simplex t0123.svg File:7-simplex t0123 A6.svg File:7-simplex t0123 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:7-simplex t0123 A4.svg File:7-simplex t0123 A3.svg File:7-simplex t0123 A2.svg
Dihedral symmetry [5] [4] [3]

Biruncicantitruncated 7-simplex

biruncicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 11760
Vertices 3360
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

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

Coordinates

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

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph File:7-simplex t1234.svg File:7-simplex t1234 A6.svg File:7-simplex t1234 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:7-simplex t1234 A4.svg File:7-simplex t1234 A3.svg File:7-simplex t1234 A2.svg
Dihedral symmetry [5] [4] [3]

Related polytopes

These polytopes are among 71 uniform 7-polytopes with A7 symmetry.

A7 polytopes
File:7-simplex t0.svg
t0 		File:7-simplex t1.svg
t1 		File:7-simplex t2.svg
t2 		File:7-simplex t3.svg
t3 		File:7-simplex t01.svg
t0,1 		File:7-simplex t02.svg
t0,2 		File:7-simplex t12.svg
t1,2 		File:7-simplex t03.svg
t0,3
File:7-simplex t13.svg
t1,3 		File:7-simplex t23.svg
t2,3 		File:7-simplex t04.svg
t0,4 		File:7-simplex t14.svg
t1,4 		File:7-simplex t24.svg
t2,4 		File:7-simplex t05.svg
t0,5 		File:7-simplex t15.svg
t1,5 		File:7-simplex t06.svg
t0,6
File:7-simplex t012.svg
t0,1,2 		File:7-simplex t013.svg
t0,1,3 		File:7-simplex t023.svg
t0,2,3 		File:7-simplex t123.svg
t1,2,3 		File:7-simplex t014.svg
t0,1,4 		File:7-simplex t024.svg
t0,2,4 		File:7-simplex t124.svg
t1,2,4 		File:7-simplex t034.svg
t0,3,4
File:7-simplex t134.svg
t1,3,4 		File:7-simplex t234.svg
t2,3,4 		File:7-simplex t015.svg
t0,1,5 		File:7-simplex t025.svg
t0,2,5 		File:7-simplex t125.svg
t1,2,5 		File:7-simplex t035.svg
t0,3,5 		File:7-simplex t135.svg
t1,3,5 		File:7-simplex t045.svg
t0,4,5
File:7-simplex t016.svg
t0,1,6 		File:7-simplex t026.svg
t0,2,6 		File:7-simplex t036.svg
t0,3,6 		File:7-simplex t0123.svg
t0,1,2,3 		File:7-simplex t0124.svg
t0,1,2,4 		File:7-simplex t0134.svg
t0,1,3,4 		File:7-simplex t0234.svg
t0,2,3,4 		File:7-simplex t1234.svg
t1,2,3,4
File:7-simplex t0125.svg
t0,1,2,5 		File:7-simplex t0135.svg
t0,1,3,5 		File:7-simplex t0235.svg
t0,2,3,5 		File:7-simplex t1235.svg
t1,2,3,5 		File:7-simplex t0145.svg
t0,1,4,5 		File:7-simplex t0245.svg
t0,2,4,5 		File:7-simplex t1245.svg
t1,2,4,5 		File:7-simplex t0345.svg
t0,3,4,5
File:7-simplex t0126.svg
t0,1,2,6 		File:7-simplex t0136.svg
t0,1,3,6 		File:7-simplex t0236.svg
t0,2,3,6 		File:7-simplex t0146.svg
t0,1,4,6 		File:7-simplex t0246.svg
t0,2,4,6 		File:7-simplex t0156.svg
t0,1,5,6 		File:7-simplex t01234.svg
t0,1,2,3,4 		File:7-simplex t01235.svg
t0,1,2,3,5
File:7-simplex t01245.svg
t0,1,2,4,5 		File:7-simplex t01345.svg
t0,1,3,4,5 		File:7-simplex t02345.svg
t0,2,3,4,5 		File:7-simplex t12345.svg
t1,2,3,4,5 		File:7-simplex t01236.svg
t0,1,2,3,6 		File:7-simplex t01246.svg
t0,1,2,4,6 		File:7-simplex t01346.svg
t0,1,3,4,6 		File:7-simplex t02346.svg
t0,2,3,4,6
File:7-simplex t01256.svg
t0,1,2,5,6 		File:7-simplex t01356.svg
t0,1,3,5,6 		File:7-simplex t012345.svg
t0,1,2,3,4,5 		File:7-simplex t012346.svg
t0,1,2,3,4,6 		File:7-simplex t012356.svg
t0,1,2,3,5,6 		File:7-simplex t012456.svg
t0,1,2,4,5,6 		File:7-simplex t0123456.svg
t0,1,2,3,4,5,6

Notes

  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

  • 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

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
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 polychoron Pentachoron 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
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