Stericated 6-orthoplexes

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6-cube t5.svg
6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png		 6-cube t15.svg
Stericated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png		 6-cube t145.svg
Steritruncated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
6-cube t135.svg
Stericantellated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png		 6-cube t1345.svg
Stericantitruncated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png		 6-cube t125.svg
Steriruncinated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
6-cube t1245.svg
Steriruncitruncated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png		 6-cube t1235.svg
Steriruncicantellated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png		 6-cube t12345.svg
Steriruncicantitruncated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex.

There are 16 unique sterications for the 6-orthoplex with permutations of truncations, cantellations, and runcinations. Eight are better represented from the stericated 6-cube.

Stericated 6-orthoplex

Stericated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol 2r2r{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 4a.pngCDel nodea.png
5-faces
4-faces
Cells
Faces
Edges 5760
Vertices 960
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Small cellated hexacontatetrapeton (Acronym: scag) (Jonathan Bowers)[1]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t15.svg 150px 150px
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 150px 150px
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 150px 150px
Dihedral symmetry [6] [4]

Steritruncated 6-orthoplex

Steritruncated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol t0,1,4{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
5-faces
4-faces
Cells
Faces
Edges 19200
Vertices 3840
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Cellitruncated hexacontatetrapeton (Acronym: catog) (Jonathan Bowers)[2]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t145.svg 150px 150px
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 150px 150px
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 150px 150px
Dihedral symmetry [6] [4]

Stericantellated 6-orthoplex

Stericantellated 6-orthoplex
Type uniform 6-polytope
Schläfli symbols t0,2,4{34,4}
rr2r{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 4a.pngCDel nodea.png
5-faces
4-faces
Cells
Faces
Edges 28800
Vertices 5760
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Cellirhombated hexacontatetrapeton (Acronym: crag) (Jonathan Bowers)[3]


Images

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t135.svg 150px 150px
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 150px 150px
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 150px 150px
Dihedral symmetry [6] [4]

Stericantitruncated 6-orthoplex

stericantitruncated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,4{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-faces
4-faces
Cells
Faces
Edges 46080
Vertices 11520
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Celligreatorhombated hexacontatetrapeton (Acronym: cagorg) (Jonathan Bowers)[4]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t1345.svg 150px 150px
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 150px 150px
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 150px 150px
Dihedral symmetry [6] [4]

Steriruncinated 6-orthoplex

steriruncinated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol t0,3,4{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
5-faces
4-faces
Cells
Faces
Edges 15360
Vertices 3840
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Celliprismated hexacontatetrapeton (Acronym: copog) (Jonathan Bowers)[5]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t125.svg 150px 150px
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 150px 150px
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 150px 150px
Dihedral symmetry [6] [4]

Steriruncitruncated 6-orthoplex

steriruncitruncated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol 2t2r{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 4a.pngCDel nodea.png
5-faces
4-faces
Cells
Faces
Edges 40320
Vertices 11520
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Celliprismatotruncated hexacontatetrapeton (Acronym: captog) (Jonathan Bowers)[6]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t1245.svg 150px 150px
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 150px 150px
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 150px 150px
Dihedral symmetry [6] [4]

Steriruncicantellated 6-orthoplex

steriruncicantellated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol t0,2,3,4{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
5-faces
4-faces
Cells
Faces
Edges 40320
Vertices 11520
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Celliprismatorhombated hexacontatetrapeton (Acronym: coprag) (Jonathan Bowers)[7]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t1235.svg 150px 150px
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 150px 150px
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 150px 150px
Dihedral symmetry [6] [4]

Steriruncicantitruncated 6-orthoplex

Steriuncicantitruncated 6-orthoplex
Type uniform 6-polytope
Schläfli symbols t0,1,2,3,4{34,4}
tr2r{3,3,3,3,4}
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 1.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 4a.pngCDel nodea.png
5-faces
4-faces
Cells
Faces
Edges 69120
Vertices 23040
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Great cellated hexacontatetrapeton (Acronym: gocog) (Jonathan Bowers)[8]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t12345.svg 150px 150px
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 150px 150px
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 150px 150px
Dihedral symmetry [6] [4]

Related polytopes

These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-orthoplex or 6-orthoplex.

<templatestyles src="Template:Hidden begin/styles.css"/>

B6 polytopes
6-cube t5.svg
β6		 6-cube t4.svg
t1β6		 6-cube t3.svg
t2β6		 6-cube t2.svg
t2γ6		 6-cube t1.svg
t1γ6		 6-cube t0.svg
γ6		 6-cube t45.svg
t0,1β6		 6-cube t35.svg
t0,2β6
6-cube t34.svg
t1,2β6		 6-cube t25.svg
t0,3β6		 6-cube t24.svg
t1,3β6		 6-cube t23.svg
t2,3γ6		 6-cube t15.svg
t0,4β6		 6-cube t14.svg
t1,4γ6		 6-cube t13.svg
t1,3γ6		 6-cube t12.svg
t1,2γ6
6-cube t05.svg
t0,5γ6		 6-cube t04.svg
t0,4γ6		 6-cube t03.svg
t0,3γ6		 6-cube t02.svg
t0,2γ6		 6-cube t01.svg
t0,1γ6		 6-cube t345.svg
t0,1,2β6		 6-cube t245.svg
t0,1,3β6		 6-cube t235.svg
t0,2,3β6
6-cube t234.svg
t1,2,3β6		 6-cube t145.svg
t0,1,4β6		 6-cube t135.svg
t0,2,4β6		 6-cube t134.svg
t1,2,4β6		 6-cube t125.svg
t0,3,4β6		 6-cube t124.svg
t1,2,4γ6		 6-cube t123.svg
t1,2,3γ6		 6-cube t045.svg
t0,1,5β6
6-cube t035.svg
t0,2,5β6		 6-cube t034.svg
t0,3,4γ6		 6-cube t025.svg
t0,2,5γ6		 6-cube t024.svg
t0,2,4γ6		 6-cube t023.svg
t0,2,3γ6		 6-cube t015.svg
t0,1,5γ6		 6-cube t014.svg
t0,1,4γ6		 6-cube t013.svg
t0,1,3γ6
6-cube t012.svg
t0,1,2γ6		 6-cube t2345.svg
t0,1,2,3β6		 6-cube t1345.svg
t0,1,2,4β6		 6-cube t1245.svg
t0,1,3,4β6		 6-cube t1235.svg
t0,2,3,4β6		 6-cube t1234.svg
t1,2,3,4γ6		 6-cube t0345.svg
t0,1,2,5β6		 6-cube t0245.svg
t0,1,3,5β6
6-cube t0235.svg
t0,2,3,5γ6		 6-cube t0234.svg
t0,2,3,4γ6		 6-cube t0145.svg
t0,1,4,5γ6		 6-cube t0135.svg
t0,1,3,5γ6		 6-cube t0134.svg
t0,1,3,4γ6		 6-cube t0125.svg
t0,1,2,5γ6		 6-cube t0124.svg
t0,1,2,4γ6		 6-cube t0123.svg
t0,1,2,3γ6
6-cube t12345.svg
t0,1,2,3,4β6		 6-cube t02345.svg
t0,1,2,3,5β6		 6-cube t01345.svg
t0,1,2,4,5β6		 6-cube t01245.svg
t0,1,2,4,5γ6		 6-cube t01235.svg
t0,1,2,3,5γ6		 6-cube t01234.svg
t0,1,2,3,4γ6		 6-cube t012345.svg
t0,1,2,3,4,5γ6


Notes

  1. ↑ Klitzing, (x3o3o3o3x4o - scag)
  2. ↑ Klitzing, (x3x3o3o3x4o - catog)
  3. ↑ Klitzing, (x3o3x3o3x4o - crag)
  4. ↑ Klitzing, (x3x3x3o3x4o - cagorg)
  5. ↑ Klitzing, (x3o3o3x3x4o - copog)
  6. ↑ Klitzing, (x3x3o3x3x4o - captog)
  7. ↑ Klitzing, (x3o3x3x3x4o - coprag)
  8. ↑ Klitzing, (x3x3x3x3x4o - gocog)

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 [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.
  • Richard Klitzing, 6D, uniform polytopes (polypeta)

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 4-polytope 5-cell 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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