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: stesa) (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: tetsa) (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: tersa) (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: togresa) (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]

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: tapsa) (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: toptasa) (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: topresa) (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: togapsa) (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: tacosa) (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: tecatsa) (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: tecresa) (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: tecgresa) (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: tecpasa) (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: tecpetsa) (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: tocpresa) (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: gotesa) (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, (x4o3o3o3o3x3o - stesa)
^ Klitzing, (x4x3o3o3o3x3o - tetsa)
^ Klitzing, (x4o3x3o3o3x3o - tersa)
^ Klitzing, (x4x3x3o3o3x3o - togresa)
^ Klitzing, (x4o3o3x3o3x3o - tapsa)
^ Klitzing, (x4x3o3x3o3x3o - toptasa)
^ Klitzing, (x4o3x3x3o3x3o - topresa)
^ Klitzing, (x4x3x3x3o3x3o - togapsa)
^ Klitzing, (x4o3o3o3x3x3o - tacosa)
^ Klitzing, (x4x3o3o3x3x3o - tecatsa)
^ Klitzing, (x4o3x3o3x3x3o - tecresa)
^ Klitzing, (x4x3x3o3x3x3o - tecgresa)
^ Klitzing, (x4o3o3x3x3x3o - tecpasa)
^ Klitzing, (x4x3o3x3x3x3o - tecpetsa)
^ Klitzing, (x4o3x3x3x3x3o - tocpresa)
^ Klitzing, (x4x3x3x3x3x3o - gotesa)

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, 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.
Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms".

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
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 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] Klitzing, (x4o3o3o3o3x3o - stesa)

[2] Klitzing, (x4x3o3o3o3x3o - tetsa)

[3] Klitzing, (x4o3x3o3o3x3o - tersa)

[4] Klitzing, (x4x3x3o3o3x3o - togresa)

[5] Klitzing, (x4o3o3x3o3x3o - tapsa)

[6] Klitzing, (x4x3o3x3o3x3o - toptasa)

[7] Klitzing, (x4o3x3x3o3x3o - topresa)

[8] Klitzing, (x4x3x3x3o3x3o - togapsa)

[9] Klitzing, (x4o3o3o3x3x3o - tacosa)

[10] Klitzing, (x4x3o3o3x3x3o - tecatsa)

[11] Klitzing, (x4o3x3o3x3x3o - tecresa)

[12] Klitzing, (x4x3x3o3x3x3o - tecgresa)

[13] Klitzing, (x4o3o3x3x3x3o - tecpasa)

[14] Klitzing, (x4x3o3x3x3x3o - tecpetsa)

[15] Klitzing, (x4o3x3x3x3x3o - tocpresa)

[16] Klitzing, (x4x3x3x3x3x3o - gotesa)

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