7-cube	Cantellated 7-cube	Bicantellated 7-cube	Tricantellated 7-cube
Birectified 7-cube	Cantitruncated 7-cube	Bicantitruncated 7-cube	Tricantitruncated 7-cube
Cantellated 7-orthoplex	Bicantellated 7-orthoplex	Cantitruncated 7-orthoplex	Bicantitruncated 7-orthoplex
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

Cantellated 7-cube
Type	uniform 7-polytope
Schläfli symbol	rr{4,3,3,3,3,3}
Coxeter diagram
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

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Bicantellated 7-cube

Bicantellated 7-cube
Type	uniform 7-polytope
Schläfli symbol	r2r{4,3,3,3,3,3}
Coxeter diagrams
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

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Tricantellated 7-cube

Tricantellated 7-cube
Type	uniform 7-polytope
Schläfli symbol	r3r{4,3,3,3,3,3}
Coxeter diagrams
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

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Cantitruncated 7-cube

Cantitruncated 7-cube
Type	uniform 7-polytope
Schläfli symbol	tr{4,3,3,3,3,3}
Coxeter diagrams
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

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

It is fifth in a series of cantitruncated hypercubes:

Petrie polygon projections

Truncated cuboctahedron	Cantitruncated tesseract	Cantitruncated 5-cube	Cantitruncated 6-cube	Cantitruncated 7-cube	Cantitruncated 8-cube

Bicantitruncated 7-cube

Bicantitruncated 7-cube
Type	uniform 7-polytope
Schläfli symbol	r2r{4,3,3,3,3,3}
Coxeter diagrams
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

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Tricantitruncated 7-cube

Tricantitruncated 7-cube
Type	uniform 7-polytope
Schläfli symbol	t3r{4,3,3,3,3,3}
Coxeter diagrams
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

Images

orthographic projections

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

Related polytopes

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

See also

• List of B7 polytopes

Notes

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
    • (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

• Polytopes of Various Dimensions
• Multi-dimensional Glossary