Runcinated 7-orthoplexes

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Orthogonal projections in B6 Coxeter plane
7-cube t6 B6.svg
7-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t36 B6.svg
Runcinated 7-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t25 B6.svg
Biruncinated 7-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t356 B6.svg
Runcitruncated 7-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t245 B6.svg
Biruncitruncated 7-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t346 B6.svg
Runcicantellated 7-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t235 B6.svg
Biruncicantellated 7-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t3456 B6.svg
Runcicantitruncated 7-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-cube t2345 B6.svg
Biruncicantitruncated 7-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

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[edit]

Runcinated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,3{35,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 23520
Vertices 2240
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Small prismated hecatonicosoctaexon (acronym: spaz) (Jonathan Bowers)[1]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t36.svg 7-cube t36 B6.svg 7-cube t36 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t36 B4.svg 7-cube t36 B3.svg 7-cube t36 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t36 A5.svg 7-cube t36 A3.svg
Dihedral symmetry [6] [4]

Biruncinated 7-orthoplex[edit]

Biruncinated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t1,4{35,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 60480
Vertices 6720
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Small biprismated hecatonicosoctaexon (Acronym sibpaz) (Jonathan Bowers)[2]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t25.svg 7-cube t25 B6.svg 7-cube t25 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t25 B4.svg 7-cube t25 B3.svg 7-cube t25 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t25 A5.svg 7-cube t25 A3.svg
Dihedral symmetry [6] [4]

Runcitruncated 7-orthoplex[edit]

Runcitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,3{35,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 50400
Vertices 6720
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Prismatotruncated hecatonicosoctaexon (acronym: potaz) (Jonathan Bowers)[3]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex 7-cube t356 B6.svg 7-cube t356 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t356 B4.svg 7-cube t356 B3.svg 7-cube t356 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t356 A5.svg 7-cube t356 A3.svg
Dihedral symmetry [6] [4]

Biruncitruncated 7-orthoplex[edit]

Biruncitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t1,2,4{35,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 120960
Vertices 20160
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Biprismatotruncated hecatonicosoctaexon (acronym: baptize) (Jonathan Bowers)[4]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t245.svg 7-cube t245 B6.svg 7-cube t245 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t245 B4.svg 7-cube t245 B3.svg 7-cube t245 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t245 A5.svg 7-cube t245 A3.svg
Dihedral symmetry [6] [4]

Runcicantellated 7-orthoplex[edit]

Runcicantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,2,3{35,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 33600
Vertices 6720
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Prismatorhombated hecatonicosoctaexon (acronym: parz) (Jonathan Bowers)[5]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex 7-cube t346 B6.svg 7-cube t346 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t346 B4.svg 7-cube t346 B3.svg 7-cube t346 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t346 A5.svg 7-cube t346 A3.svg
Dihedral symmetry [6] [4]

Biruncicantellated 7-orthoplex[edit]

biruncicantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t1,3,4{35,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 100800
Vertices 20160
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Biprismatorhombated hecatonicosoctaexon (acronym: boparz) (Jonathan Bowers)[6]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t235.svg 7-cube t235 B6.svg 7-cube t235 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t235 B4.svg 7-cube t235 B3.svg 7-cube t235 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t235 A5.svg 7-cube t235 A3.svg
Dihedral symmetry [6] [4]

Runcicantitruncated 7-orthoplex[edit]

Runcicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3{35,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 60480
Vertices 13440
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Great prismated hecatonicosoctaexon (acronym: gopaz) (Jonathan Bowers)[7]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex 7-cube t3456 B6.svg 7-cube t3456 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t3456 B4.svg 7-cube t3456 B3.svg 7-cube t3456 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t3456 A5.svg 7-cube t3456 A3.svg
Dihedral symmetry [6] [4]

Biruncicantitruncated 7-orthoplex[edit]

biruncicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t1,2,3,4{35,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 161280
Vertices 40320
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Great biprismated hecatonicosoctaexon (acronym: gibpaz) (Jonathan Bowers)[8]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t1234.svg 7-cube t1234 B6.svg 7-cube t1234 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t1234 B4.svg 7-cube t1234 B3.svg 7-cube t1234 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t1234 A5.svg 7-cube t1234 A3.svg
Dihedral symmetry [6] [4]

Notes[edit]

  1. ^ Klitzing, (o3o3o3x3o3o4x - spaz)
  2. ^ Klitzing, (o3x3o3o3x3o4o - sibpaz)
  3. ^ Klitzing, (o3o3o3x3x3o4x - potaz)
  4. ^ Klitzing, (o3o3x3o3x3x4o - baptize)
  5. ^ Klitzing, (o3o3o3x3x3o4x - parz)
  6. ^ Klitzing, (o3x3o3x3x3o4o - boparz)
  7. ^ Klitzing, (o3o3o3x3x3x4x - gopaz)
  8. ^ Klitzing, (o3o3x3x3x3x3o - gibpaz)

References[edit]

  • 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. (1966)
  • Richard Klitzing, 7D, uniform polytopes (polyexa) o3o3o3x3o3o4x - spaz, o3x3o3o3x3o4o - sibpaz, o3o3o3x3x3o4x - potaz, o3o3x3o3x3x4o - baptize, o3o3o3x3x3o4x - parz, o3x3o3x3x3o4o - boparz, o3o3o3x3x3x4x - gopaz, o3o3x3x3x3x3o - gibpaz

External links[edit]