Hexicated 7-orthoplexes

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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

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.

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[edit]

Hexitruncated 7-orthoplex
Type Uniform 7-polytope
Schläfli symbol t0,1,6{35,4
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 29568
Vertices 5376
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Petitruncated heptacross

Images[edit]

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[edit]

Hexicantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,2,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 94080
Vertices 13440
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Petirhombated heptacross

Images[edit]

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[edit]

Hexicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 134400
Vertices 26880
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Petigreatorhombated heptacross

Images[edit]

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[edit]

Hexiruncitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,3,6{35,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 322560
Vertices 53760
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Petiprismatotruncated heptacross

Images[edit]

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[edit]

Hexiruncicantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,2,3,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 268800
Vertices 53760
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

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

Alternate names[edit]

  • Petiprismatorhombated heptacross

Images[edit]

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[edit]

hexisteritruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 322560
Vertices 53760
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Peticellitruncated heptacross

Images[edit]

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[edit]

Hexiruncicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 483840
Vertices 107520
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Petigreatoprismated heptacross

Images[edit]

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[edit]

Hexistericantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 806400
Vertices 161280
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Peticelligreatorhombated heptacross

Images[edit]

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[edit]

Hexisteriruncitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,3,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 725760
Vertices 161280
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Peticelliprismatotruncated heptacross

Images[edit]

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[edit]

hexipenticantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,5,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 483840
Vertices 107520
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Petiterigreatorhombated heptacross

Images[edit]

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[edit]

Hexisteriruncicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 1290240
Vertices 322560
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Great petacellated heptacross

Images[edit]

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[edit]

Hexipentiruncicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3,5,6{35,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 1290240
Vertices 322560
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names[edit]

  • Petiterigreatoprismated heptacross

Images[edit]

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[edit]

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 [1]
      • (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)".

External links[edit]

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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