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	t0,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,1,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,2,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,1,2,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,3,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,1,3,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,2,3,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,1,2,3,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,4,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,1,4,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,2,4,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,1,2,4,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,3,4,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,1,3,4,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	40320
Vertices	10080
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,2,3,4,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	40320
Vertices	10080
Vertex figure
Coxeter groups	B7, [4,3^5]
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	t0,1,2,3,4,5{4,3^5}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B7, [4,3^5]
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

1. Klitzing, (x3o3o3o3o3x4o - )
2. Klitzing, (x3x3o3o3o3x4o - )
3. Klitzing, (x3o3x3o3o3x4o - )
4. Klitzing, (x3x3x3oxo3x4o - )
5. Klitzing, (x3o3o3x3o3x4o - )
6. Klitzing, (x3x3o3x3o3x4o - )
7. Klitzing, (x3o3x3x3o3x4o - )
8. Klitzing, (x3x3x3x3o3x4o - )
9. Klitzing, (x3o3o3o3x3x4o - )
10. Klitzing, (x3x3o3o3x3x4o - )
11. Klitzing, (x3o3x3o3x3x4o - )
12. Klitzing, (x3x3x3o3x3x4o - )
13. Klitzing, (x3o3o3x3x3x4o - )
14. Klitzing, (x3x3o3x3x3x4o - )
15. Klitzing, (x3o3x3x3x3x4o - )
16. 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.
• Klitzing, Richard. "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". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / E9 / E10 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds