Runcinated 7-cubes

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Orthogonal projections in BC₇ Coxeter plane
7-cube	Runcinated 7-cube	Biruncinated 7-cube
Runcitruncated 7-cube	Biruncitruncated 7-cube	Runcicantellated 7-cube
Biruncicantellated 7-cube	Runcicantitruncated 7-cube	Biruncicantitruncated 7-cube

In seven-dimensional geometry, a runcinated 7-cube is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-cube.

There are 16 unique runcinations of the 7-cube with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-orthoplex.

These polytopes are among 127 uniform 7-polytopes with BC₇ symmetry.

Runcinated 7-cube

Runcinated 7-cube
Type	uniform polyexon
Schläfli symbol	t_0,3{4,3⁵}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	33600
Vertices	4480
Vertex figure
Coxeter groups	BC₇, [4,3⁵]
Properties	convex

Alternate names

Small prismated hepteract (acronym: spesa) (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]

Biruncinated 7-cube

Biruncinated 7-cube
Type	uniform polyexon
Schläfli symbol	t_1,4{4,3⁵}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	67200
Vertices	8960
Vertex figure
Coxeter groups	BC₇, [4,3⁵]
Properties	convex

Alternate names

Small biprismated hepteract (Acronym sibposa) (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]

Runcitruncated 7-cube

Runcitruncated 7-cube
Type	uniform polyexon
Schläfli symbol	t_0,1,3{4,3⁵}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	73920
Vertices	13440
Vertex figure
Coxeter groups	BC₇, [4,3⁵]
Properties	convex

Alternate names

Prismatotruncated hepteract (acronym: petsa) (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]

Biruncitruncated 7-cube

Biruncitruncated 7-cube
Type	uniform polyexon
Schläfli symbol	t_1,2,4{4,3⁵}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	134400
Vertices	26880
Vertex figure
Coxeter groups	BC₇, [4,3⁵]
Properties	convex

Alternate names

Biprismatotruncated hepteract (acronym: biptesa) (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]

Runcicantellated 7-cube

Runcicantellated 7-cube
Type	uniform polyexon
Schläfli symbol	t_0,2,3{4,3⁵}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	53760
Vertices	13440
Vertex figure
Coxeter groups	BC₇, [4,3⁵]
Properties	convex

Alternate names

Prismatorhombated hepteract (acronym: parsa) (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]

Biruncicantellated 7-cube

biruncicantellated 7-cube
Type	uniform polyexon
Schläfli symbol	t_1,3,4{4,3⁵}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	120960
Vertices	26880
Vertex figure
Coxeter groups	BC₇, [4,3⁵]
Properties	convex

Alternate names

Biprismatorhombated hepteract (acronym: bopresa) (Jonathan Bowers)^[6]

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]

Runcicantitruncated 7-cube

Runcicantitruncated 7-cube
Type	uniform polyexon
Schläfli symbol	t_0,1,2,3{4,3⁵}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	94080
Vertices	26880
Vertex figure
Coxeter groups	BC₇, [4,3⁵]
Properties	convex

Alternate names

Great prismated hepteract (acronym: gapsa) (Jonathan Bowers)^[7]

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]

Biruncicantitruncated 7-cube

biruncicantitruncated 7-cube
Type	uniform polyexon
Schläfli symbol	t_1,2,3,4{4,3⁵}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	188160
Vertices	53760
Vertex figure
Coxeter groups	BC₇, [4,3⁵]
Properties	convex

Alternate names

Great biprismated hepteract (acronym: gibposa) (Jonathan Bowers)^[8]

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]

Notes

↑ Klitzing, (x3o3o3x3o3o4o - spesa)
↑ Klitzing, (o3x3o3o3x3o4o - sibposa)
↑ Klitzing, (x3x3o3x3o3o4o - petsa)
↑ Klitzing, (o3x3x3o3x3o4o - biptesa)
↑ Klitzing, (x3o3x3x3o3o4o - parsa)
↑ Klitzing, (o3o3x3x3o3x4o - bopresa)
↑ Klitzing, (x3x3x3x3o3o4o - gapsa)
↑ Klitzing, (o3x3x3x3x3o3o - gibposa)

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, editied 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. (1966)
Richard Klitzing, 7D, uniform polytopes (polyexa) x3o3o3x3o3o4o - spo, o3x3o3o3x3o4o - sibpo, x3x3o3x3o3o4o - patto, o3x3x3o3x3o4o - bipto, x3o3x3x3o3o4o - paro, x3x3x3x3o3o4o - gapo, o3x3x3x3x3o3o- gibpo

External links

Olshevsky, George, Cross polytope at Glossary for Hyperspace.
Polytopes of Various Dimensions
Multi-dimensional Glossary

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	BC_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	5-cell	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

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