Orthogonal projections in A6 Coxeter plane
6-simplexRuncinated 6-simplexBiruncinated 6-simplex
Runcitruncated 6-simplexBiruncitruncated 6-simplexRuncicantellated 6-simplex
Runcicantitruncated 6-simplexBiruncicantitruncated 6-simplex

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
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)

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 planeA6A5A4
Graph
Dihedral symmetry[7][6][5]
Ak Coxeter planeA3A2
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)

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 planeA6A5A4
Graph
Symmetry[[7]](*)=[14][6][[5]](*)=[10]
Ak Coxeter planeA3A2
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)

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 planeA6A5A4
Graph
Dihedral symmetry[7][6][5]
Ak Coxeter planeA3A2
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)

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 planeA6A5A4
Graph
Dihedral symmetry[7][6][5]
Ak Coxeter planeA3A2
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)

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 planeA6A5A4
Graph
Dihedral symmetry[7][6][5]
Ak Coxeter planeA3A2
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)

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 planeA6A5A4
Graph
Dihedral symmetry[7][6][5]
Ak Coxeter planeA3A2
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)

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 planeA6A5A4
Graph
Symmetry[[7]](*)=[14][6][[5]](*)=[10]
Ak Coxeter planeA3A2
Graph
Symmetry[4][[3]](*)=[6]

Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

Related uniform 6-polytopes

The runcinated 6-simplexes are in a set 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
t0t1t2t0,1t0,2t1,2t0,3t1,3t2,3
t0,4t1,4t0,5t0,1,2t0,1,3t0,2,3t1,2,3t0,1,4t0,2,4
t1,2,4t0,3,4t0,1,5t0,2,5t0,1,2,3t0,1,2,4t0,1,3,4t0,2,3,4t1,2,3,4
t0,1,2,5t0,1,3,5t0,2,3,5t0,1,4,5t0,1,2,3,4t0,1,2,3,5t0,1,2,4,5t0,1,2,3,4,5

Notes

  • 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 Ivić 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. . x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof

External links

vteFundamental convex regular and uniform polytopes in dimensions 2–10
FamilyAnBnI2(p) / DnE6 / E7 / E8 / F4 / G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
Uniform polyhedronTetrahedronOctahedron • CubeDemicubeDodecahedron • Icosahedron
Uniform polychoronPentachoron16-cell • TesseractDemitesseract24-cell120-cell • 600-cell
Uniform 5-polytope5-simplex5-orthoplex • 5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex • 6-cube6-demicube122 • 221
Uniform 7-polytope7-simplex7-orthoplex • 7-cube7-demicube132 • 231 • 321
Uniform 8-polytope8-simplex8-orthoplex • 8-cube8-demicube142 • 241 • 421
Uniform 9-polytope9-simplex9-orthoplex • 9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex • 10-cube10-demicube
Uniform n-polytopen-simplexn-orthoplex • n-cuben-demicube1k2 • 2k1 • k21n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations