Hexicated 7-orthoplexes

In seven-dimensional geometry, a hexicated 7-orthoplex (also hexicated 7-cube) is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-orthoplex.

Orthogonal projections in B4 Coxeter plane

7-orthoplex
Hexicated 7-orthoplex
Hexicated 7-cube
Hexi-truncated 7-orthoplex
Hexi-cantellated 7-orthoplex
Hexicanti-truncated 7-orthoplex

Hexirunci-truncated 7-orthoplex
Hexirunci-cantellated 7-orthoplex
Hexisteri-truncated 7-orthoplex
Hexiruncicanti-truncated 7-orthoplex
Hexistericanti-truncated 7-orthoplex

Hexisterirunci-truncated 7-orthoplex
Hexipenticanti-truncated 7-orthoplex
Hexisteriruncicanti-truncated 7-orthoplex
Hexipentiruncicanti-truncated 7-orthoplex

There are 32 hexications for the 7-orthoplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. 12 are represented here, while 20 are more easily constructed from the 7-cube.

Hexitruncated 7-orthoplex
Hexitruncated 7-orthoplex
TypeUniform 7-polytope
Schläfli symbolt0,1,6{35,4
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges29568
Vertices5376
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Petitruncated heptacross

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]

Hexicantellated 7-orthoplex
Hexicantellated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,2,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges94080
Vertices13440
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Petirhombated heptacross

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]

Hexicantitruncated 7-orthoplex
Hexicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges134400
Vertices26880
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Petigreatorhombated heptacross

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]

Hexiruncitruncated 7-orthoplex
Hexiruncitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,3,6{35,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges322560
Vertices53760
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Petiprismatotruncated heptacross

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]

Hexiruncicantellated 7-orthoplex
Hexiruncicantellated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,2,3,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges268800
Vertices53760
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

In seven-dimensional geometry, a hexiruncicantellated 7-orthoplex is a uniform 7-polytope.

Alternate names
  • Petiprismatorhombated heptacross

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]

Hexisteritruncated 7-orthoplex
hexisteritruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges322560
Vertices53760
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Peticellitruncated heptacross

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]

Hexiruncicantitruncated 7-orthoplex
Hexiruncicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges483840
Vertices107520
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Petigreatoprismated heptacross

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]

Hexistericantitruncated 7-orthoplex
Hexistericantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges806400
Vertices161280
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Peticelligreatorhombated heptacross

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]

Hexisteriruncitruncated 7-orthoplex
Hexisteriruncitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,3,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges725760
Vertices161280
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Peticelliprismatotruncated heptacross

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]

Hexipenticantitruncated 7-orthoplex
hexipenticantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,5,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges483840
Vertices107520
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Petiterigreatorhombated heptacross

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]

Hexisteriruncicantitruncated 7-orthoplex
Hexisteriruncicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges1290240
Vertices322560
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Great petacellated heptacross

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]

Hexipentiruncicantitruncated 7-orthoplex
Hexipentiruncicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,5,6{35,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges1290240
Vertices322560
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names
  • Petiterigreatoprismated heptacross

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]

Notes

    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
        • (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, PhD (1966)
    • Klitzing, Richard. "7D uniform polytopes (polyexa)".
    Fundamental convex regular and uniform polytopes in dimensions 2–10
    Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
    Regular polygon Triangle Square p-gon Hexagon Pentagon
    Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
    Uniform polychoron Pentachoron 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
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