Orthogonal projections in B6 Coxeter plane
7-orthoplexRuncinated 7-orthoplexBiruncinated 7-orthoplex
Runcitruncated 7-orthoplexBiruncitruncated 7-orthoplexRuncicantellated 7-orthoplex
Biruncicantellated 7-orthoplexRuncicantitruncated 7-orthoplexBiruncicantitruncated 7-orthoplex

In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-orthoplex.

There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-cube.

These polytopes are among 127 uniform 7-polytopes with B7 symmetry.

Runcinated 7-orthoplex

Runcinated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges23520
Vertices2240
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Small prismated hecatonicosaoctaexon (acronym: spaz) (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]

Biruncinated 7-orthoplex

Biruncinated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt1,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges60480
Vertices6720
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Small biprismated hecatonicosaoctaexon (acronym: sibpaz) (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]

Runcitruncated 7-orthoplex

Runcitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges50400
Vertices6720
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Prismatotruncated hecatonicosaoctaexon (acronym: potaz) (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]

Biruncitruncated 7-orthoplex

Biruncitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt1,2,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges120960
Vertices20160
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Biprismatotruncated hecatonicosaoctaexon (acronym: baptize) (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]

Runcicantellated 7-orthoplex

Runcicantellated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,2,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges33600
Vertices6720
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Prismatorhombated hecatonicosaoctaexon (acronym: parz) (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]

Biruncicantellated 7-orthoplex

Biruncicantellated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt1,3,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges100800
Vertices20160
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Biprismatorhombated hecatonicosoctaexon (acronym: boparz) (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]

Runcicantitruncated 7-orthoplex

Runcicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges60480
Vertices13440
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Great prismated hecatonicosaoctaexon (acronym: gopaz) (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]

Biruncicantitruncated 7-orthoplex

Biruncicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt1,2,3,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges161280
Vertices40320
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Great biprismated hecatonicosaoctaexon (acronym: gibpaz) (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]

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. (1966)
  • Klitzing, Richard. . x3o3o3x3o3o4o - spaz, o3x3o3o3x3o4o - sibpaz, x3x3o3x3o3o4o - potaz, o3x3x3o3x3o4o - baptize, x3o3x3x3o3o4o - parz, o3x3o3x3x3o4o - boparz, x3x3x3x3o3o4o - gopaz, o3x3x3x3x3o4o - gibpaz

External links

vteFundamental convex regular and uniform polytopes in dimensions 2–10
FamilyAnBnI2(p) / DnE6 / E7 / E8 / F4 / G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
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