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{4,35}
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 hepteract (acronym:) (Jonathan Bowers)

Pentitruncated 7-cube

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

Penticantellated 7-cube

Penticantellated 7-cube
Type	uniform 7-polytope
Schläfli symbol	t0,2,5{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)

Penticantitruncated 7-cube

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

Pentiruncinated 7-cube

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

Pentiruncitruncated 7-cube

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

Pentiruncicantellated 7-cube

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

Pentiruncicantitruncated 7-cube

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

Pentistericated 7-cube

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

Pentisteritruncated 7-cube

pentisteritruncated 7-cube
Type	uniform 7-polytope
Schläfli symbol	t0,1,4,5{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)

Pentistericantellated 7-cube

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

Pentistericantitruncated 7-cube

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

Pentisteriruncinated 7-cube

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

Pentisteriruncitruncated 7-cube

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

Pentisteriruncicantellated 7-cube

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

Pentisteriruncicantitruncated 7-cube

pentisteriruncicantitruncated 7-cube
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,4,5{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 hepteract (acronym:) (Jonathan Bowers)

Related polytopes

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

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)". https://bendwavy.org/klitzing/dimensions/polyexa.htm.  x3o3o3o3o3x4o, x3x3o3o3o3x4o, x3o3x3o3o3x4o, x3x3x3oxo3x4o, x3o3o3x3o3x4o, x3x3o3x3o3x4o, x3o3x3x3o3x4o, x3x3x3x3o3x4o, x3o3o3o3x3x4o, x3x3o3o3x3x4o, x3o3x3o3x3x4o, x3x3x3o3x3x4o, x3o3o3x3x3x4o, x3x3o3x3x3x4o, x3o3x3x3x3x4o, x3x3x3x3x3x3o

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary