Runcinated 6-simplexes

From Wikipedia, the free encyclopedia
  (Redirected from Runcicantitruncated 6-simplex)
Jump to: navigation, search
6-simplex t0.svg
6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t03.svg
Runcinated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t14.svg
Biruncinated 6-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t013.svg
Runcitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t124.svg
Biruncitruncated 6-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t023.svg
Runcicantellated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t0123.svg
Runcicantitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t1234.svg
Biruncicantitruncated 6-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel 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[edit]

Runcinated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,3{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

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

Coordinates[edit]

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[edit]

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

Biruncinated 6-simplex[edit]

biruncinated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel 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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t14.svg 6-simplex t14 A5.svg 6-simplex t14 A4.svg
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph 6-simplex t14 A3.svg 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[edit]

Runcitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,3{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

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

Coordinates[edit]

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[edit]

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

Biruncitruncated 6-simplex[edit]

biruncitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel 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]

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

Coordinates[edit]

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]

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

Runcicantellated 6-simplex[edit]

Runcicantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

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

Coordinates[edit]

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]

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

Runcicantitruncated 6-simplex[edit]

Runcicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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]

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

Coordinates[edit]

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]

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

Biruncicantitruncated 6-simplex[edit]

biruncicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel 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]

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

Coordinates[edit]

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]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t1234.svg 6-simplex t1234 A5.svg 6-simplex t1234 A4.svg
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph 6-simplex t1234 A3.svg 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.


Related uniform 6-polytopes[edit]

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.

Notes[edit]

  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[edit]

  • 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. "6D uniform polytopes (polypeta)".  x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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