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Runcinated 6-simplexes

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6-simplex
Runcinated 6-simplex
Biruncinated 6-simplex

Runcitruncated 6-simplex
Biruncitruncated 6-simplex
Runcicantellated 6-simplex

Runcicantitruncated 6-simplex
Biruncicantitruncated 6-simplex
Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex.

There are 8 unique runcinations of the 6-simplex with permutations of truncations, and cantellations.

Runcinated 6-simplex

Runcinated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 70
4-faces 455
Cells 1330
Faces 1610
Edges 840
Vertices 140
Vertex figure
Coxeter group A6, , order 5040
Properties convex

Alternate names

  • Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry

Biruncinated 6-simplex

biruncinated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 84
4-faces 714
Cells 2100
Faces 2520
Edges 1260
Vertices 210
Vertex figure
Coxeter group A6, ], order 10080
Properties convex

Alternate names

  • Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Symmetry ]= ]=
Ak Coxeter plane A3 A2
Graph
Symmetry ]=
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.


Runcitruncated 6-simplex

Runcitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 70
4-faces 560
Cells 1820
Faces 2800
Edges 1890
Vertices 420
Vertex figure
Coxeter group A6, , order 5040
Properties convex

Alternate names

  • Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry

Biruncitruncated 6-simplex

biruncitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 84
4-faces 714
Cells 2310
Faces 3570
Edges 2520
Vertices 630
Vertex figure
Coxeter group A6, , order 5040
Properties convex

Alternate names

  • Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry

Runcicantellated 6-simplex

Runcicantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 70
4-faces 455
Cells 1295
Faces 1960
Edges 1470
Vertices 420
Vertex figure
Coxeter group A6, , order 5040
Properties convex

Alternate names

  • Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry

Runcicantitruncated 6-simplex

Runcicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 70
4-faces 560
Cells 1820
Faces 3010
Edges 2520
Vertices 840
Vertex figure
Coxeter group A6, , order 5040
Properties convex

Alternate names

  • Runcicantitruncated heptapeton
  • Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry

Biruncicantitruncated 6-simplex

biruncicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 84
4-faces 714
Cells 2520
Faces 4410
Edges 3780
Vertices 1260
Vertex figure
Coxeter group A6, ], order 10080
Properties convex

Alternate names

  • Biruncicantitruncated heptapeton
  • Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Symmetry ]= ]=
Ak Coxeter plane A3 A2
Graph
Symmetry ]=
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.


Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

A6 polytopes

t0
t1
t2
t0,1
t0,2
t1,2
t0,3
t1,3
t2,3

t0,4
t1,4
t0,5
t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4
t0,2,4

t1,2,4
t0,3,4
t0,1,5
t0,2,5
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
t0,1,4,5
t0,1,2,3,4
t0,1,2,3,5
t0,1,2,4,5
t0,1,2,3,4,5

Notes

  1. Klitzing, (x3o3o3x3o3o - spil)
  2. Klitzing, (o3x3o3o3x3o - sibpof)
  3. Klitzing, (x3x3o3x3o3o - patal)
  4. Klitzing, (o3x3x3o3x3o - bapril)
  5. Klitzing, (x3o3x3x3o3o - pril)
  6. Klitzing, (x3x3x3x3o3o - gapil)
  7. Klitzing, (o3x3x3x3x3o - gibpof)

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
      • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I,
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
  • Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof

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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