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Pentellated 7-simplexes

From Wikipedia, the free encyclopedia


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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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

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

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

Coordinates

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

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]]

Related polytopes

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

  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

  • 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

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
This page was last edited on 26 December 2017, at 04:34
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