Orthogonal projections in B6 Coxeter plane
7-cubeCantellated 7-cubeBicantellated 7-cubeTricantellated 7-cube
Birectified 7-cubeCantitruncated 7-cubeBicantitruncated 7-cubeTricantitruncated 7-cube
Cantellated 7-orthoplexBicantellated 7-orthoplexCantitruncated 7-orthoplexBicantitruncated 7-orthoplex

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

Cantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolrr{4,3,3,3,3,3}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges16128
Vertices2688
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

Alternate names

  • Small rhombated hepteract (acronym: sersa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Bicantellated 7-cube

Bicantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolr2r{4,3,3,3,3,3}
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges40320
Vertices6720
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

Alternate names

  • Small birhombated hepteract (acronym: sibrosa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Tricantellated 7-cube

Tricantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolr3r{4,3,3,3,3,3}
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges47040
Vertices6720
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

Alternate names

  • Small trirhombihepteractihecatonicosaoctaexon (acronym: strasaz) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Cantitruncated 7-cube

Cantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symboltr{4,3,3,3,3,3}
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges18816
Vertices5376
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

Alternate names

  • Great rhombated hepteract (acronym: gersa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

It is fifth in a series of cantitruncated hypercubes:

Petrie polygon projections
Truncated cuboctahedronCantitruncated tesseractCantitruncated 5-cubeCantitruncated 6-cubeCantitruncated 7-cubeCantitruncated 8-cube

Bicantitruncated 7-cube

Bicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolr2r{4,3,3,3,3,3}
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges47040
Vertices13440
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

Alternate names

  • Great birhombated hepteract (acronym: gibrosa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Tricantitruncated 7-cube

Tricantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt3r{4,3,3,3,3,3}
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges53760
Vertices13440
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

Alternate names

  • Great trirhombihepteractihecatonicosaoctaexon (acronym: gotrasaz) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graphtoo complex
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Related polytopes

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

See also

Notes

  • 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 Ivić Weiss, Wiley-Interscience Publication, 1995, , ISBN 978-0-471-01003-6 (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. . x4o3x3o3o3o3o - sersa, o4x3o3x3o3o3o - sibrosa, o4o3x3o3x3o3o - strasaz, x4x3x3o3o3o3o - gersa, o4x3x3x3o3o3o - gibrosa, o4o3x3x3x3o3o - gotrasaz

External links

vteFundamental convex regular and uniform polytopes in dimensions 2–10
FamilyAnBnI2(p) / DnE6 / E7 / E8 / F4 / G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
Uniform polyhedronTetrahedronOctahedron • CubeDemicubeDodecahedron • Icosahedron
Uniform polychoronPentachoron16-cell • TesseractDemitesseract24-cell120-cell • 600-cell
Uniform 5-polytope5-simplex5-orthoplex • 5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex • 6-cube6-demicube122 • 221
Uniform 7-polytope7-simplex7-orthoplex • 7-cube7-demicube132 • 231 • 321
Uniform 8-polytope8-simplex8-orthoplex • 8-cube8-demicube142 • 241 • 421
Uniform 9-polytope9-simplex9-orthoplex • 9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex • 10-cube10-demicube
Uniform n-polytopen-simplexn-orthoplex • n-cuben-demicube1k2 • 2k1 • k21n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations