Runcinated 7-cubes

Orthogonal projections in B₆ 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 B₇ symmetry.

Runcinated 7-cube

Runcinated 7-cube
Type	uniform 7-polytope
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	B₇, [4,3⁵]
Properties	convex

Alternate names

Small prismated hepteract (acronym: spesa) (Jonathan Bowers)

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 7-polytope
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	B₇, [4,3⁵]
Properties	convex

Alternate names

Small biprismated hepteract (Acronym sibposa) (Jonathan Bowers)

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 7-polytope
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	B₇, [4,3⁵]
Properties	convex

Alternate names

Prismatotruncated hepteract (acronym: petsa) (Jonathan Bowers)

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 7-polytope
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	B₇, [4,3⁵]
Properties	convex

Alternate names

Biprismatotruncated hepteract (acronym: biptesa) (Jonathan Bowers)

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 7-polytope
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	B₇, [4,3⁵]
Properties	convex

Alternate names

Prismatorhombated hepteract (acronym: parsa) (Jonathan Bowers)

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 7-polytope
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	B₇, [4,3⁵]
Properties	convex

Alternate names

Biprismatorhombated hepteract (acronym: bopresa) (Jonathan Bowers)

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 7-polytope
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	B₇, [4,3⁵]
Properties	convex

Alternate names

Great prismated hepteract (acronym: gapsa) (Jonathan Bowers)

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 7-polytope
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	B₇, [4,3⁵]
Properties	convex

Alternate names

Great biprismated hepteract (acronym: gibposa) (Jonathan Bowers)

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, (x4o3o3x3o3o3o - spesa)
↑ Klitzing, (o4x3o3o3x3o3o - sibposa)
↑ Klitzing, (x4x3o3x3o3o3o - petsa)
↑ Klitzing, (o4x3x3o3x3o3o - biptesa)
↑ Klitzing, (x4o3x3x3o3o3o - parsa)
↑ Klitzing, (o4x3o3x3x3o3o - bopresa)
↑ Klitzing, (x4x3x3x3o3o3o - gapsa)
↑ Klitzing, (o4x3x3x3x3o3o - 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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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)
Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms". x4o3o3x3o3o3o - spesa, o4x3o3o3x3o3o - sibposa, x4x3o3x3o3o3o - petsa, o4x3x3o3x3o3o - biptesa, x4o3x3x3o3o3o - parsa, o4x3o3x3x3o3o - bopresa, x4x3x3x3o3o3o - gapsa, o4x3x3x3x3o3o - gibposa

External links

v t e Fundamental convex regular and uniform polytopes
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
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 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 • Polytope operations

[1] Klitzing, (x4o3o3x3o3o3o - spesa)

[2] Klitzing, (o4x3o3o3x3o3o - sibposa)

[3] Klitzing, (x4x3o3x3o3o3o - petsa)

[4] Klitzing, (o4x3x3o3x3o3o - biptesa)

[5] Klitzing, (x4o3x3x3o3o3o - parsa)

[6] Klitzing, (o4x3o3x3x3o3o - bopresa)

[7] Klitzing, (x4x3x3x3o3o3o - gapsa)

[8] Klitzing, (o4x3x3x3x3o3o - gibposa)