Runcinated 6-simplexes

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.


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

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

Runcinated 6-simplex

Runcinated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces70
4-faces455
Cells1330
Faces1610
Edges840
Vertices140
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

  • Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)[1]

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 [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Biruncinated 6-simplex

biruncinated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces84
4-faces714
Cells2100
Faces2520
Edges1260
Vertices210
Vertex figure
Coxeter groupA6, [[35]], order 10080
Propertiesconvex

Alternate names

  • Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)[2]

Coordinates

The vertices of the biruncinted 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 [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

Runcitruncated 6-simplex

Runcitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces70
4-faces560
Cells1820
Faces2800
Edges1890
Vertices420
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

  • Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers)[3]

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 [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Biruncitruncated 6-simplex

biruncitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces84
4-faces714
Cells2310
Faces3570
Edges2520
Vertices630
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

  • Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)[4]

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 [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Runcicantellated 6-simplex

Runcicantellated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces70
4-faces455
Cells1295
Faces1960
Edges1470
Vertices420
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

  • Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)[5]

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 [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Runcicantitruncated 6-simplex

Runcicantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces70
4-faces560
Cells1820
Faces3010
Edges2520
Vertices840
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

  • Runcicantitruncated heptapeton
  • Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)[6]

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 [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]

Biruncicantitruncated 6-simplex

biruncicantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces84
4-faces714
Cells2520
Faces4410
Edges3780
Vertices1260
Vertex figure
Coxeter groupA6, [[35]], order 10080
Propertiesconvex

Alternate names

  • Biruncicantitruncated heptapeton
  • Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)[7]

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 [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] 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, [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. "6D uniform polytopes (polypeta)". x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof
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
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