Pentellated 7-orthoplexes

Orthogonal projections in B₆ Coxeter plane
7-orthoplex	Pentellated 7-orthoplex	Pentitruncated 7-orthoplex	Penticantellated 7-orthoplex
Penticantitruncated 7-orthoplex	Pentiruncinated 7-orthoplex	Pentiruncitruncated 7-orthoplex	Pentiruncicantellated 7-orthoplex
Pentiruncicantitruncated 7-orthoplex	Pentistericated 7-orthoplex	Pentisteritruncated 7-orthoplex	Pentistericantellated 7-orthoplex
Pentistericantitruncated 7-orthoplex	Pentisteriruncinated 7-orthoplex	Pentisteriruncitruncated 7-orthoplex	Pentisteriruncicantellated 7-orthoplex
Pentisteriruncicantitruncated 7-orthoplex

In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-orthoplex.

There are 32 unique pentellations of the 7-orthoplex with permutations of truncations, cantellations, runcinations, and sterications. 16 are more simply constructed relative to the 7-cube.

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

Pentellated 7-orthoplex

Pentellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	20160
Vertices	2688
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Small terated hecatonicosoctaexon (acronym: Staz) (Jonathan Bowers)^[1]

Coordinates

Coordinates are permutations of (0,1,1,1,1,1,2)√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]

Pentitruncated 7-orthoplex

pentitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	87360
Vertices	13440
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Teritruncated hecatonicosoctaexon (acronym: Tetaz) (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]

Coordinates

Coordinates are permutations of (0,1,1,1,1,2,3).

Penticantellated 7-orthoplex

Penticantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	188160
Vertices	26880
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Terirhombated hecatonicosoctaexon (acronym: Teroz) (Jonathan Bowers)^[3]

Coordinates

Coordinates are permutations of (0,1,1,1,2,2,3)√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]

Penticantitruncated 7-orthoplex

penticantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	295680
Vertices	53760
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Terigreatorhombated hecatonicosoctaexon (acronym: Tograz) (Jonathan Bowers)^[4]

Coordinates

Coordinates are permutations of (0,1,1,1,2,3,4)√2.

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-orthoplex

pentiruncinated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,3,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	174720
Vertices	26880
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Teriprismated hecatonicosoctaexon (acronym: Topaz) (Jonathan Bowers)^[5]

Coordinates

The coordinates are permutations of (0,1,1,2,2,2,3)√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]

Pentiruncitruncated 7-orthoplex

pentiruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	443520
Vertices	80640
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Teriprismatotruncated hecatonicosoctaexon (acronym: Toptaz) (Jonathan Bowers)^[6]

Coordinates

Coordinates are permutations of (0,1,1,2,2,3,4)√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]

Pentiruncicantellated 7-orthoplex

pentiruncicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,3,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	403200
Vertices	80640
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Teriprismatorhombated hecatonicosoctaexon (acronym: Toparz) (Jonathan Bowers)^[7]

Coordinates

Coordinates are permutations of (0,1,1,2,3,3,4)√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]

Pentiruncicantitruncated 7-orthoplex

pentiruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	725760
Vertices	161280
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Terigreatoprismated hecatonicosoctaexon (acronym: Tegopaz) (Jonathan Bowers)^[8]

Coordinates

Coordinates are permutations of (0,1,1,2,3,4,5)√2.

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]

Pentistericated 7-orthoplex

pentistericated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,4,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	67200
Vertices	13440
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Tericellated hecatonicosoctaexon (acronym: Tocaz) (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]

Coordinates

Coordinates are permutations of (0,1,2,2,2,2,3)√2.

Pentisteritruncated 7-orthoplex

pentisteritruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,4,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	241920
Vertices	53760
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Tericellitruncated hecatonicosoctaexon (acronym: Tacotaz) (Jonathan Bowers)^[10]

Coordinates

Coordinates are permutations of (0,1,2,2,2,3,4)√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]

Pentistericantellated 7-orthoplex

pentistericantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,4,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	403200
Vertices	80640
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Tericellirhombated hecatonicosoctaexon (acronym: Tocarz) (Jonathan Bowers)^[11]

Coordinates

Coordinates are permutations of (0,1,2,2,3,3,4)√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]

Pentistericantitruncated 7-orthoplex

pentistericantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,4,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	645120
Vertices	161280
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Tericelligreatorhombated hecatonicosoctaexon (acronym: Tecagraz) (Jonathan Bowers)^[12]

Coordinates

Coordinates are permutations of (0,1,2,2,3,4,5)√2.

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-orthoplex

Pentisteriruncinated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,3,4,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	241920
Vertices	53760
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

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

Coordinates

Coordinates are permutations of (0,1,2,3,3,3,4)√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]

Pentisteriruncitruncated 7-orthoplex

pentisteriruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3,4,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	645120
Vertices	161280
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Tericelliprismatotruncated hecatonicosoctaexon (acronym: Tecpotaz) (Jonathan Bowers)^[14]

Coordinates

Coordinates are permutations of (0,1,2,3,3,4,5)√2.

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-orthoplex

pentisteriruncicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,3,4,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	645120
Vertices	161280
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

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

Coordinates

Coordinates are permutations of (0,1,2,3,4,4,5)√2.

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-orthoplex

pentisteriruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3,4,5{3⁵,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	1128960
Vertices	322560
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Great terated hecatonicosoctaexon (acronym: Gotaz) (Jonathan Bowers)^[16]

Coordinates

Coordinates are permutations of (0,1,2,3,4,5,6)√2.

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]

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
  - (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)

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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