Cantellated 7-cubes

Orthogonal projections in B₆ Coxeter plane
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

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

Alternate names

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

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

Alternate names

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

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

Alternate names

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

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

Alternate names

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

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

Alternate names

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

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

Alternate names

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

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.

Notes

^ Klitizing, (x3o3x3o3o3o4o - sersa)
^ Klitizing, (o3x3o3x3o3o4o - sibrosa)
^ Klitizing, (o3o3x3o3x3o4o - strasaz)
^ Klitizing, (x3x3x3o3o3o4o - gersa)
^ Klitizing, (o3x3x3x3o3o4o - gibrosa)
^ Klitizing, (o3o3x3x3x3o4o - gotrasaz)

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

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[1] Klitizing, (x3o3x3o3o3o4o - sersa)

[2] Klitizing, (o3x3o3x3o3o4o - sibrosa)

[3] Klitizing, (o3o3x3o3x3o4o - strasaz)

[4] Klitizing, (x3x3x3o3o3o4o - gersa)

[5] Klitizing, (o3x3x3x3o3o4o - gibrosa)

[6] Klitizing, (o3o3x3x3x3o4o - gotrasaz)

[1]

[2]

[3]

[4]

[5]

[6]

Cantellated 7-cubes

Contents

Cantellated 7-cube

Alternate names

Images

Bicantellated 7-cube

Alternate names

Images

Tricantellated 7-cube

Alternate names

Images

Cantitruncated 7-cube

Alternate names

Images

Bicantitruncated 7-cube

Alternate names

Images

Tricantitruncated 7-cube

Alternate names

Images

Related polytopes

See also

Notes

References

External links

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