Runcinated 7-orthoplexes

Orthogonal projections in B₆ Coxeter plane
7-orthoplex	Runcinated 7-orthoplex	Biruncinated 7-orthoplex
Runcitruncated 7-orthoplex	Biruncitruncated 7-orthoplex	Runcicantellated 7-orthoplex
Biruncicantellated 7-orthoplex	Runcicantitruncated 7-orthoplex	Biruncicantitruncated 7-orthoplex

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

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

These polytopes are among 127 uniform 7-polytopes with B₇ symmetry.

Runcinated 7-orthoplex

Runcinated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,3{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	23520
Vertices	2240
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Small prismated hecatonicosoctaexon (acronym: spaz) (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-orthoplex

Biruncinated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_1,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	60480
Vertices	6720
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Small biprismated hecatonicosoctaexon (Acronym sibpaz) (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-orthoplex

Runcitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	50400
Vertices	6720
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Prismatotruncated hecatonicosoctaexon (acronym: potaz) (Jonathan Bowers)^[3]

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]

Biruncitruncated 7-orthoplex

Biruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_1,2,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	120960
Vertices	20160
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Biprismatotruncated hecatonicosoctaexon (acronym: baptize) (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-orthoplex

Runcicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,3{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	33600
Vertices	6720
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Prismatorhombated hecatonicosoctaexon (acronym: parz) (Jonathan Bowers)^[5]

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]

Biruncicantellated 7-orthoplex

biruncicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_1,3,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	100800
Vertices	20160
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Biprismatorhombated hecatonicosoctaexon (acronym: boparz) (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-orthoplex

Runcicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	60480
Vertices	13440
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Great prismated hecatonicosoctaexon (acronym: gopaz) (Jonathan Bowers)^[7]

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]

Biruncicantitruncated 7-orthoplex

biruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_1,2,3,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	161280
Vertices	40320
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Great biprismated hecatonicosoctaexon (acronym: gibpaz) (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, (o3o3o3x3o3o4x - spaz)
↑ Klitzing, (o3x3o3o3x3o4o - sibpaz)
↑ Klitzing, (o3o3o3x3x3o4x - potaz)
↑ Klitzing, (o3o3x3o3x3x4o - baptize)
↑ Klitzing, (o3o3o3x3x3o4x - parz)
↑ Klitzing, (o3x3o3x3x3o4o - boparz)
↑ Klitzing, (o3o3o3x3x3x4x - gopaz)
↑ Klitzing, (o3o3x3x3x3x3o - gibpaz)

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 Wiley: Kaleidoscopes: Selected Writings of H.S.M. Coxeter
  - (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)". o3o3o3x3o3o4x - spaz, o3x3o3o3x3o4o - sibpaz, o3o3o3x3x3o4x - potaz, o3o3x3o3x3x4o - baptize, o3o3o3x3x3o4x - parz, o3x3o3x3x3o4o - boparz, o3o3o3x3x3x4x - gopaz, o3o3x3x3x3x3o - gibpaz

External links

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

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[1] Klitzing, (o3o3o3x3o3o4x - spaz)

[2] Klitzing, (o3x3o3o3x3o4o - sibpaz)

[3] Klitzing, (o3o3o3x3x3o4x - potaz)

[4] Klitzing, (o3o3x3o3x3x4o - baptize)

[5] Klitzing, (o3o3o3x3x3o4x - parz)

[6] Klitzing, (o3x3o3x3x3o4o - boparz)

[7] Klitzing, (o3o3o3x3x3x4x - gopaz)

[8] Klitzing, (o3o3x3x3x3x3o - gibpaz)

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