Stericated 7-orthoplexes

Orthogonal projections in B₆ Coxeter plane
7-orthoplex	Stericated 7-orthoplex	Steritruncated 7-orthoplex
Bisteritruncated 7-orthoplex	Stericantellated 7-orthoplex	Stericantitruncated 7-orthoplex
Bistericantitruncated 7-orthoplex	Steriruncinated 7-orthoplex	Steriruncitruncated 7-orthoplex
Steriruncicantellated 7-orthoplex	Bisteriruncitruncated 7-orthoplex	Steriruncicantitruncated 7-orthoplex

In seven-dimensional geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-orthoplex.

There are 24 unique sterication for the 7-orthoplex with permutations of truncations, cantellations, and runcinations. 14 are more simply constructed from the 7-cube.

This polytope is one of 127 uniform 7-polytopes with B₇ symmetry.

Stericated 7-orthoplex

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

Alternate names

Small cellated hecatonicosoctaexon (acronym: ) (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]

Steritruncated 7-orthoplex

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

Alternate names

Cellitruncated hecatonicosoctaexon (acronym: ) (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]

Bisteritruncated 7-orthoplex

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

Alternate names

Bicellitruncated hecatonicosoctaexon (acronym: ) (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]

Stericantellated 7-orthoplex

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

Alternate names

Cellirhombated hecatonicosoctaexon (acronym: ) (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]

Stericantitruncated 7-orthoplex

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

Alternate names

Celligreatorhombated hecatonicosoctaexon (acronym: ) (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]

Bistericantitruncated 7-orthoplex

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

Alternate names

Bicelligreatorhombated hecatonicosoctaexon (acronym: ) (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]

Steriruncinated 7-orthoplex

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

Alternate names

Celliprismated hecatonicosoctaexon (acronym: ) (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]

Steriruncitruncated 7-orthoplex

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

Alternate names

Celliprismatotruncated hecatonicosoctaexon (acronym: ) (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]

Steriruncicantellated 7-orthoplex

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

Alternate names

Celliprismatorhombated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[9]

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]

Steriruncicantitruncated 7-orthoplex

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

Alternate names

Great cellated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)^[10]

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

^ Klitizing, (x3o3o3o3x3o4o - )
^ Klitizing, (x3x3o3o3x3o4o - )
^ Klitizing, (o3x3x3o3o3x4o - )
^ Klitizing, (x3o3x3o3x3o4o - )
^ Klitizing, (x3x3x3o3x3o4o - )
^ Klitizing, (o3x3x3x3o3x4o - )
^ Klitizing, (x3o3o3x3x3o4o - )
^ Klitizing, (x3x3x3o3x3o4o - )
^ Klitizing, (x3o3x3x3x3o4o - )
^ Klitizing, (x3x3x3x3x3o4o - )

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)".

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, (x3o3o3o3x3o4o - )

[2] Klitizing, (x3x3o3o3x3o4o - )

[3] Klitizing, (o3x3x3o3o3x4o - )

[4] Klitizing, (x3o3x3o3x3o4o - )

[5] Klitizing, (x3x3x3o3x3o4o - )

[6] Klitizing, (o3x3x3x3o3x4o - )

[7] Klitizing, (x3o3o3x3x3o4o - )

[8] Klitizing, (x3x3x3o3x3o4o - )

[9] Klitizing, (x3o3x3x3x3o4o - )

[10] Klitizing, (x3x3x3x3x3o4o - )

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