Runcinated 6-simplexes

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6-simplex
File: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:6-simplex t03.svg
Runcinated 6-simplex
File:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:6-simplex t14.svg
Biruncinated 6-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.png
File:6-simplex t013.svg
Runcitruncated 6-simplex
File:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:6-simplex t124.svg
Biruncitruncated 6-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.png 		File:6-simplex t023.svg
Runcicantellated 6-simplex
File:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:6-simplex t0123.svg
Runcicantitruncated 6-simplex
File:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:6-simplex t1234.svg
Biruncicantitruncated 6-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.png
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

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Runcinated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,3{3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
5-faces 70
4-faces 455
Cells 1330
Faces 1610
Edges 840
Vertices 140
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names

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

Coordinates

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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

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

Biruncinated 6-simplex

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biruncinated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.png
5-faces 84
4-faces 714
Cells 2100
Faces 2520
Edges 1260
Vertices 210
Vertex figure
Coxeter group A6, [[35]], order 10080
Properties convex

Alternate names

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

Coordinates

[edit | edit source]

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

[edit | edit source]
orthographic projections
Ak Coxeter plane A6 A5 A4
Graph File:6-simplex t14.svg File:6-simplex t14 A5.svg File:6-simplex t14 A4.svg
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph File:6-simplex t14 A3.svg File:6-simplex t14 A2.svg
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.


Runcitruncated 6-simplex

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Runcitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,3{3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
5-faces 70
4-faces 560
Cells 1820
Faces 2800
Edges 1890
Vertices 420
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names

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

Coordinates

[edit | edit source]

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

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

Biruncitruncated 6-simplex

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biruncitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.png
5-faces 84
4-faces 714
Cells 2310
Faces 3570
Edges 2520
Vertices 630
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names

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

Coordinates

[edit | edit source]

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

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

Runcicantellated 6-simplex

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Runcicantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
5-faces 70
4-faces 455
Cells 1295
Faces 1960
Edges 1470
Vertices 420
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names

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

Coordinates

[edit | edit source]

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

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

Runcicantitruncated 6-simplex

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Runcicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
5-faces 70
4-faces 560
Cells 1820
Faces 3010
Edges 2520
Vertices 840
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names

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

Coordinates

[edit | edit source]

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

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

Biruncicantitruncated 6-simplex

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biruncicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel 3.pngFile:CDel node.png
5-faces 84
4-faces 714
Cells 2520
Faces 4410
Edges 3780
Vertices 1260
Vertex figure
Coxeter group A6, [[35]], order 10080
Properties convex

Alternate names

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

Coordinates

[edit | edit source]

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

[edit | edit source]
orthographic projections
Ak Coxeter plane A6 A5 A4
Graph File:6-simplex t1234.svg File:6-simplex t1234 A5.svg File:6-simplex t1234 A4.svg
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph File:6-simplex t1234 A3.svg File:6-simplex t1234 A2.svg
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.


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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
Error creating thumbnail:
t0 		File:6-simplex t1.svg
t1 		File:6-simplex t2.svg
t2 		File:6-simplex t01.svg
t0,1 		File:6-simplex t02.svg
t0,2 		File:6-simplex t12.svg
t1,2 		File:6-simplex t03.svg
t0,3 		File:6-simplex t13.svg
t1,3 		File:6-simplex t23.svg
t2,3
File:6-simplex t04.svg
t0,4 		File:6-simplex t14.svg
t1,4 		File:6-simplex t05.svg
t0,5 		File:6-simplex t012.svg
t0,1,2 		File:6-simplex t013.svg
t0,1,3 		File:6-simplex t023.svg
t0,2,3 		File:6-simplex t123.svg
t1,2,3 		File:6-simplex t014.svg
t0,1,4 		File:6-simplex t024.svg
t0,2,4
File:6-simplex t124.svg
t1,2,4 		File:6-simplex t034.svg
t0,3,4 		File:6-simplex t015.svg
t0,1,5 		File:6-simplex t025.svg
t0,2,5 		File:6-simplex t0123.svg
t0,1,2,3 		File:6-simplex t0124.svg
t0,1,2,4 		File:6-simplex t0134.svg
t0,1,3,4 		File:6-simplex t0234.svg
t0,2,3,4 		File:6-simplex t1234.svg
t1,2,3,4
File:6-simplex t0125.svg
t0,1,2,5 		File:6-simplex t0135.svg
t0,1,3,5 		File:6-simplex t0235.svg
t0,2,3,5 		File:6-simplex t0145.svg
t0,1,4,5 		File:6-simplex t01234.svg
t0,1,2,3,4 		File:6-simplex t01235.svg
t0,1,2,3,5 		File:6-simplex t01245.svg
t0,1,2,4,5 		File:6-simplex t012345.svg
t0,1,2,3,4,5

Notes

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  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

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  • 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, wiley.com, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
      • (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.
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof
[edit | edit source]
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 • Polytope operations
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