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-cubePentellated 7-cubePentitruncated 7-cubePenticantellated 7-cube
Penticantitruncated 7-cubePentiruncinated 7-cubePentiruncitruncated 7-cubePentiruncicantellated 7-cube
Pentiruncicantitruncated 7-cubePentistericated 7-cubePentisteritruncated 7-cubePentistericantellated 7-cube
Pentistericantitruncated 7-cubePentisteriruncinated 7-cubePentisteriruncitruncated 7-cubePentisteriruncicantellated 7-cube
Pentisteriruncicantitruncated 7-cube

Pentellated 7-cube

Pentellated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Small terated hepteract (acronym: stesa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentitruncated 7-cube

Pentitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,1,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Teritruncated hepteract (acronym: tetsa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Penticantellated 7-cube

Penticantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,2,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Terirhombated hepteract (acronym: tersa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Penticantitruncated 7-cube

Penticantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,1,2,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Terigreatorhombated hepteract (acronym: togresa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentiruncinated 7-cube

Pentiruncinated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,3,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Teriprismated hepteract (acronym: tapsa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentiruncitruncated 7-cube

Pentiruncitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,1,3,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Teriprismatotruncated hepteract (acronym: toptosa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentiruncicantellated 7-cube

Pentiruncicantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,2,3,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Teriprismatorhombated hepteract (acronym: topresa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentiruncicantitruncated 7-cube

Pentiruncicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Terigreatoprismated hepteract (acronym: togapsa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graphtoo complex
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graphtoo complextoo complex
Dihedral symmetry[6][4]

Pentistericated 7-cube

Pentistericated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,4,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Tericellated hepteract (acronym: tacosa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentisteritruncated 7-cube

Pentisteritruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,1,4,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Tericellitruncated hepteract (acronym: tecatsa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentistericantellated 7-cube

Pentistericantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,2,4,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Tericellirhombated hepteract (acronym: tecresa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentistericantitruncated 7-cube

Pentistericantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,1,2,4,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Tericelligreatorhombated hepteract (acronym: tecgresa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graphtoo complex
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentisteriruncinated 7-cube

Pentisteriruncinated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,3,4,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Bipenticantitruncated 7-cube as t1,2,3,6{4,35}
  • Tericelliprismated hepteract (acronym: tecpasa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graph
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentisteriruncitruncated 7-cube

Pentisteriruncitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,1,3,4,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges40320
Vertices10080
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Tericelliprismatotruncated hepteract (acronym: tecpetsa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graphtoo complex
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentisteriruncicantellated 7-cube

Pentisteriruncicantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,2,3,4,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges40320
Vertices10080
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Bipentiruncicantitruncated 7-cube as t1,2,3,4,6{4,35}
  • Tericelliprismatorhombated hepteract (acronym: tocpresa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graphtoo complex
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Pentisteriruncicantitruncated 7-cube

Pentisteriruncicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,4,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Great terated hepteract (acronym: gotesa) (Jonathan Bowers)

Images

Orthographic projections
Coxeter planeB7 / A6B6 / D7B5 / D6 / A4
Graphtoo complex
Dihedral symmetry[14][12][10]
Coxeter planeB4 / D5B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[8][6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Related polytopes

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

Notes

  • 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 Ivić 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.
  • Klitzing, Richard. .

External links

vteFundamental convex regular and uniform polytopes in dimensions 2–10
FamilyAnBnI2(p) / DnE6 / E7 / E8 / F4 / G2Hn
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Uniform polyhedronTetrahedronOctahedron • CubeDemicubeDodecahedron • Icosahedron
Uniform polychoronPentachoron16-cell • TesseractDemitesseract24-cell120-cell • 600-cell
Uniform 5-polytope5-simplex5-orthoplex • 5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex • 6-cube6-demicube122 • 221
Uniform 7-polytope7-simplex7-orthoplex • 7-cube7-demicube132 • 231 • 321
Uniform 8-polytope8-simplex8-orthoplex • 8-cube8-demicube142 • 241 • 421
Uniform 9-polytope9-simplex9-orthoplex • 9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex • 10-cube10-demicube
Uniform n-polytopen-simplexn-orthoplex • n-cuben-demicube1k2 • 2k1 • k21n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations