Cantellated 7-orthoplexes

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7-cube t6 B6.svg
7-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t46 B6.svg
Cantellated 7-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t35 B6.svg
Bicantellated 7-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t3 B6.svg
Birectified 7-orthoplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t456 B6.svg
Cantitruncated 7-orthoplex
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.pngCDel 4.pngCDel node.png
7-cube t345 B6.svg
Bicantitruncated 7-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t02 B6.svg
Cantellated 7-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-cube t13 B6.svg
Bicantellated 7-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-cube t24 B6.svg
Tricantellated 7-cube
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-cube t012 B6.svg
Cantitruncated 7-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-cube t123 B6.svg
Bicantitruncated 7-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-cube t234 B6.svg
Tricantitruncated 7-cube
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Orthogonal projections in B6 Coxeter plane

In seven-dimensional geometry, a cantellated 7-orthoplex is a convex uniform 7-polytope, being a cantellation of the regular 7-orthoplex.

There are ten degrees of cantellation for the 7-orthoplex, including truncations. Six are most simply constructible from the dual 7-cube.

Cantellated 7-orthoplex[edit]

Cantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol rr{3,3,3,3,3,4}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 7560
Vertices 840
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names[edit]

  • Small rhombated hecatonicosoctaexon (acronym: sarz) (Jonathan Bowers)[1]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t46.svg 7-cube t46 B6.svg 7-cube t46 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t46 B4.svg 7-cube t46 B3.svg 7-cube t46 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t46 A5.svg 7-cube t46 A3.svg
Dihedral symmetry [6] [4]

Bicantellated 7-orthoplex[edit]

Bicantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol 2rr{3,3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names[edit]

  • Small birhombated hecatonicosoctaexon (acronym: sebraz) (Jonathan Bowers)[2]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t35.svg 7-cube t35 B6.svg 7-cube t35 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t35 B4.svg 7-cube t35 B3.svg 7-cube t35 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t35 A5.svg 7-cube t35 A3.svg
Dihedral symmetry [6] [4]

Cantitruncated 7-orthoplex[edit]

Cantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol tr{3,3,3,3,3,4}
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.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 8400
Vertices 1680
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names[edit]

  • Great rhombated hecatonicosoctaexon (acronym: garz) (Jonathan Bowers)[3]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex 7-cube t456 B6.svg 7-cube t456 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t456 B4.svg 7-cube t456 B3.svg 7-cube t456 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t456 A5.svg 7-cube t456 A3.svg
Dihedral symmetry [6] [4]

Bicantitruncated 7-orthoplex[edit]

Bicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol 2tr{3,3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 30240
Vertices 6720
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names[edit]

  • Great birhombated hecatonicosoctaexon (acronym: gebraz) (Jonathan Bowers)[4]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex 7-cube t345 B6.svg 7-cube t345 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t345 B4.svg 7-cube t345 B3.svg 7-cube t345 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t345 A5.svg 7-cube t345 A3.svg
Dihedral symmetry [6] [4]

Related polytopes[edit]

These polytopes are from a family of 127 uniform 7-polytopes with B7 symmetry.

See also[edit]

Notes[edit]

  1. ^ Klitizing, (o3o3o3o3x3o4x - sarz)
  2. ^ Klitizing, (o3o3o3x3o3x4o - sebraz)
  3. ^ Klitizing, (o3o3o3o3x3x4x - garz)
  4. ^ Klitizing, (o3o3o3x3x3x4o - gebraz)

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. 1966
  • Klitzing, Richard. "7D uniform polytopes (polyexa)".  - o3o3o3o3x3o4x - sarz, o3o3o3x3o3x4o - sebraz, o3o3o3o3x3x4x - garz, o3o3o3x3x3x4o - gebraz

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