Pentellated 7-cubes

In seven-dimensional geometry, a pentellated 7-cube is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-cube. There are 32 unique pentellations of the 7-cube with permutations of truncations, cantellations, runcinations, and sterications. 16 are more simply constructed relative to the 7-orthoplex.

7-cube	Pentellated 7-cube	Pentitruncated 7-cube	Penticantellated 7-cube
Penticantitruncated 7-cube	Pentiruncinated 7-cube	Pentiruncitruncated 7-cube	Pentiruncicantellated 7-cube
Pentiruncicantitruncated 7-cube	Pentistericated 7-cube	Pentisteritruncated 7-cube	Pentistericantellated 7-cube
Pentistericantitruncated 7-cube	Pentisteriruncinated 7-cube	Pentisteriruncitruncated 7-cube	Pentisteriruncicantellated 7-cube
Pentisteriruncicantitruncated 7-cube

Pentellated 7-cube

Pentellated 7-cube
Type	uniform 7-polytope
Schläfli symbol	t_0,5{4,3⁵}
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 terated hepteract (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]

Pentitruncated 7-cube

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

Alternate names

Teritruncated hepteract (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]

Penticantellated 7-cube

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

Alternate names

Terirhombated hepteract (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]

Penticantitruncated 7-cube

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

Alternate names

Terigreatorhombated hepteract (acronym: ) (Jonathan Bowers)^[4]

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]

Pentiruncinated 7-cube

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

Alternate names

Teriprismated hepteract (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]

Pentiruncitruncated 7-cube

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

Alternate names

Teriprismatotruncated hepteract (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]

Pentiruncicantellated 7-cube

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

Alternate names

Teriprismatorhombated hepteract (acronym: ) (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]

Pentiruncicantitruncated 7-cube

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

Alternate names

Terigreatoprismated hepteract (acronym: ) (Jonathan Bowers)^[8]

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	too complex	too complex
Dihedral symmetry	[6]	[4]

Pentistericated 7-cube

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

Alternate names

Tericellated hepteract (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]

Pentisteritruncated 7-cube

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

Alternate names

Tericellitruncated hepteract (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]

Pentistericantellated 7-cube

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

Alternate names

Tericellirhombated hepteract (acronym: ) (Jonathan Bowers)^[11]

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]

Pentistericantitruncated 7-cube

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

Alternate names

Tericelligreatorhombated hepteract (acronym: ) (Jonathan Bowers)^[12]

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]

Pentisteriruncinated 7-cube

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

Alternate names

Bipenticantitruncated 7-cube as t_1,2,3,6{4,3⁵}
Tericelliprismated hepteract (acronym: ) (Jonathan Bowers)^[13]

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]

Pentisteriruncitruncated 7-cube

pentisteriruncitruncated 7-cube
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3,4,5{4,3⁵}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	40320
Vertices	10080
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Tericelliprismatotruncated hepteract (acronym: ) (Jonathan Bowers)^[14]

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]

Pentisteriruncicantellated 7-cube

pentisteriruncicantellated 7-cube
Type	uniform 7-polytope
Schläfli symbol	t_0,2,3,4,5{4,3⁵}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	40320
Vertices	10080
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Bipentiruncicantitruncated 7-cube as t_1,2,3,4,6{4,3⁵}
Tericelliprismatorhombated hepteract (acronym: ) (Jonathan Bowers)^[15]

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]

Pentisteriruncicantitruncated 7-cube

pentisteriruncicantitruncated 7-cube
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3,4,5{4,3⁵}
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 terated hepteract (acronym: ) (Jonathan Bowers)^[16]

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 a part of a set of 127 uniform 7-polytopes with B₇ symmetry.

Notes

↑ Klitzing, (x3o3o3o3o3x4o - )
↑ Klitzing, (x3x3o3o3o3x4o - )
↑ Klitzing, (x3o3x3o3o3x4o - )
↑ Klitzing, (x3x3x3oxo3x4o - )
↑ Klitzing, (x3o3o3x3o3x4o - )
↑ Klitzing, (x3x3o3x3o3x4o - )
↑ Klitzing, (x3o3x3x3o3x4o - )
↑ Klitzing, (x3x3x3x3o3x4o - )
↑ Klitzing, (x3o3o3o3x3x4o - )
↑ Klitzing, (x3x3o3o3x3x4o - )
↑ Klitzing, (x3o3x3o3x3x4o - )
↑ Klitzing, (x3x3x3o3x3x4o - )
↑ Klitzing, (x3o3o3x3x3x4o - )
↑ Klitzing, (x3x3o3x3x3x4o - )
↑ Klitzing, (x3o3x3x3x3x4o - )
↑ Klitzing, (x3x3x3x3x3x4o - )

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.
Richard Klitzing, 7D, uniform polytopes (polyexa) x3o3o3o3o3x4o, x3x3o3o3o3x4o, x3o3x3o3o3x4o, x3x3x3oxo3x4o, x3o3o3x3o3x4o, x3x3o3x3o3x4o, x3o3x3x3o3x4o, x3x3x3x3o3x4o, x3o3o3o3x3x4o, x3x3o3o3x3x4o, x3o3x3o3x3x4o, x3x3x3o3x3x4o, x3o3o3x3x3x4o, x3x3o3x3x3x4o, x3o3x3x3x3x4o, x3x3x3x3x3x3o

External links

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

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 4-polytope	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 and compounds

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