Pentellated 7-orthoplexes

Orthogonal projections in B6 Coxeter plane
7-orthoplex	120px Pentellated 7-orthoplex	120px Pentitruncated 7-orthoplex	120px Penticantellated 7-orthoplex
120px Penticantitruncated 7-orthoplex	120px Pentiruncinated 7-orthoplex	120px Pentiruncitruncated 7-orthoplex	120px Pentiruncicantellated 7-orthoplex
120px Pentiruncicantitruncated 7-orthoplex	120px Pentistericated 7-orthoplex	120px Pentisteritruncated 7-orthoplex	120px Pentistericantellated 7-orthoplex
120px Pentistericantitruncated 7-orthoplex	120px Pentisteriruncinated 7-orthoplex	120px Pentisteriruncitruncated 7-orthoplex	120px Pentisteriruncicantellated 7-orthoplex
120px 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	t0,5{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)

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	150px	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentitruncated 7-orthoplex

pentitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,5{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)

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph	150px	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
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	t0,2,5{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)

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	150px	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Penticantitruncated 7-orthoplex

penticantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,5{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)

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	150px	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentiruncinated 7-orthoplex

pentiruncinated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,3,5{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)

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	150px	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentiruncitruncated 7-orthoplex

pentiruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,5{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)

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	150px	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentiruncicantellated 7-orthoplex

pentiruncicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,3,5{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)

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	180px	180px	180px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	180px	180px	180px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	180px	180px
Dihedral symmetry	[6]	[4]

Pentiruncicantitruncated 7-orthoplex

pentiruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,5{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)

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	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentistericated 7-orthoplex

pentistericated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,4,5{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)

Images

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph	150px	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
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	t0,1,4,5{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)

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	150px	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentistericantellated 7-orthoplex

pentistericantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,4,5{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)

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	150px	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentistericantitruncated 7-orthoplex

pentistericantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,4,5{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)

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	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentisteriruncinated 7-orthoplex

Pentisteriruncinated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,3,4,5{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)

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	150px	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentisteriruncitruncated 7-orthoplex

pentisteriruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,4,5{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)

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	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentisteriruncicantellated 7-orthoplex

pentisteriruncicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,3,4,5{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)

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	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentisteriruncicantitruncated 7-orthoplex

pentisteriruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,4,5{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)

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	150px	150px
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

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.
• Richard Klitzing, 7D, uniform polytopes (polyexa)

External links

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