Cantellated 7-cubes

From Wikipedia, the free encyclopedia
  (Redirected from Cantellated 7-cube)
Jump to: navigation, search
7-cube t0 B6.svg
7-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.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 t2 B6.svg
Birectified 7-cube
CDel node.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 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
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 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
Orthogonal projections in B6 Coxeter plane

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

There are 10 degrees of cantellation for the 7-cube, including truncations. 4 are most simply constructible from the dual 7-orthoplex.

Cantellated 7-cube[edit]

Cantellated 7-cube
Type uniform 7-polytope
Schläfli symbol rr{4,3,3,3,3,3}
Coxeter diagram 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
6-faces
5-faces
4-faces
Cells
Faces
Edges 16128
Vertices 2688
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names[edit]

  • Small rhombated hepteract (acronym: sersa) (Jonathan Bowers)[1]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t02.svg 7-cube t02 B6.svg 7-cube t02 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t02 B4.svg 7-cube t02 B3.svg 7-cube t02 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t02 A5.svg 7-cube t02 A3.svg
Dihedral symmetry [6] [4]

Bicantellated 7-cube[edit]

Bicantellated 7-cube
Type uniform 7-polytope
Schläfli symbol r2r{4,3,3,3,3,3}
Coxeter diagrams 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
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 40320
Vertices 6720
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names[edit]

  • Small birhombated hepteract (acronym: sibrosa) (Jonathan Bowers)[2]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t13.svg 7-cube t13 B6.svg 7-cube t13 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t13 B4.svg 7-cube t13 B3.svg 7-cube t13 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t13 A5.svg 7-cube t13 A3.svg
Dihedral symmetry [6] [4]

Tricantellated 7-cube[edit]

Tricantellated 7-cube
Type uniform 7-polytope
Schläfli symbol r3r{4,3,3,3,3,3}
Coxeter diagrams 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
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 47040
Vertices 6720
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names[edit]

  • Small trirhombihepteractihecatonicosoctaexon (acronym: strasaz) (Jonathan Bowers)[3]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t24.svg 7-cube t24 B6.svg 7-cube t24 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t24 B4.svg 7-cube t24 B3.svg 7-cube t24 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t24 A5.svg 7-cube t24 A3.svg
Dihedral symmetry [6] [4]

Cantitruncated 7-cube[edit]

Cantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol tr{4,3,3,3,3,3}
Coxeter diagrams 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
6-faces
5-faces
4-faces
Cells
Faces
Edges 18816
Vertices 5376
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names[edit]

  • Great rhombated hepteract (acronym: gersa) (Jonathan Bowers)[4]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t012.svg 7-cube t012 B6.svg 7-cube t012 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t012 B4.svg 7-cube t012 B3.svg 7-cube t012 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t012 A5.svg 7-cube t012 A3.svg
Dihedral symmetry [6] [4]

Bicantitruncated 7-cube[edit]

Bicantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol r2r{4,3,3,3,3,3}
Coxeter diagrams 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
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 47040
Vertices 13440
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names[edit]

  • Great birhombated hepteract (acronym: gibrosa) (Jonathan Bowers)[5]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t123.svg 7-cube t123 B6.svg 7-cube t123 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t123 B4.svg 7-cube t123 B3.svg 7-cube t123 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t123 A5.svg 7-cube t123 A3.svg
Dihedral symmetry [6] [4]

Tricantitruncated 7-cube[edit]

Tricantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t3r{4,3,3,3,3,3}
Coxeter diagrams 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
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 53760
Vertices 13440
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names[edit]

  • Great trirhombihepteractihecatonicosoctaexon (acronym: gotrasaz) (Jonathan Bowers)[6]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex 7-cube t234 B6.svg 7-cube t234 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t234 B4.svg 7-cube t234 B3.svg 7-cube t234 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t234 A5.svg 7-cube t234 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, (x3o3x3o3o3o4o - sersa)
  2. ^ Klitizing, (o3x3o3x3o3o4o - sibrosa)
  3. ^ Klitizing, (o3o3x3o3x3o4o - strasaz)
  4. ^ Klitizing, (x3x3x3o3o3o4o - gersa)
  5. ^ Klitizing, (o3x3x3x3o3o4o - gibrosa)
  6. ^ Klitizing, (o3o3x3x3x3o4o - gotrasaz)

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. "7D uniform polytopes (polyexa)".  x3o3x3o3o3o4o- sersa, o3x3o3x3o3o4o - sibrosa, o3o3x3o3x3o4o - strasaz, x3x3x3o3o3o4o - gersa, o3x3x3x3o3o4o - gibrosa, o3o3x3x3x3o4o - gotrasaz

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