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Hexicated 7-cubes

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(Redirected from Hexicantitruncated 7-cube) Shape in seven - dimensional geometry
Orthogonal projections in B4 Coxeter plane

7-cube
Hexicated 7-cube
Hexitruncated 7-cube
Hexicantellated 7-cube

Hexiruncinated 7-cube
Hexicantitruncated 7-cube
Hexiruncitruncated 7-cube
Hexiruncicantellated 7-cube

Hexisteritruncated 7-cube
Hexistericantellated 7-cube
Hexipentitruncated 7-cube
Hexiruncicantitruncated 7-cube

Hexistericantitruncated 7-cube
Hexisteriruncitruncated 7-cube
Hexisteriruncicantellated 7-cube
Hexipenticantitruncated 7-cube

Hexipentiruncitruncated 7-cube
Hexisteriruncicantitruncated 7-cube
Hexipentiruncicantitruncated 7-cube
Hexipentistericantitruncated 7-cube

Hexipentisteriruncicantitruncated 7-cube
(Omnitruncated 7-cube)

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

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

The simple hexicated 7-cube is also called an expanded 7-cube, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-cube. The highest form, the hexipentisteriruncicantitruncated 7-cube is more simply called a omnitruncated 7-cube with all of the nodes ringed.

These polytope are among a family of 127 uniform 7-polytopes with B7 symmetry.

Hexicated 7-cube

Hexicated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-cube, or alternately can be seen as an expansion operation.

Alternate names

  • Small petated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexitruncated 7-cube

hexitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Petitruncated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexicantellated 7-cube

Hexicantellated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,2,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Petirhombated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexiruncinated 7-cube

Hexiruncinated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,3,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Petiprismated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexicantitruncated 7-cube

Hexicantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,2,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Petigreatorhombated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexiruncitruncated 7-cube

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

Alternate names

  • Petiprismatotruncated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexiruncicantellated 7-cube

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

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

Alternate names

  • Petiprismatorhombated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexisteritruncated 7-cube

hexisteritruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,4,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Peticellitruncated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexistericantellated 7-cube

hexistericantellated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,2,4,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Peticellirhombihepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexipentitruncated 7-cube

Hexipentitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,5,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Petiteritruncated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexiruncicantitruncated 7-cube

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

Alternate names

  • Petigreatoprismated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph too complex too complex
Dihedral symmetry

Hexistericantitruncated 7-cube

Hexistericantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,2,4,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Peticelligreatorhombated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexisteriruncitruncated 7-cube

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

Alternate names

  • Peticelliprismatotruncated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexisteriruncicantellated 7-cube

Hexisteriruncitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,2,3,4,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Peticelliprismatorhombihepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexipenticantitruncated 7-cube

hexipenticantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,2,5,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Petiterigreatorhombated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexipentiruncitruncated 7-cube

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

Alternate names

  • Great petacellated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexisteriruncicantitruncated 7-cube

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

Alternate names

  • Great petacellated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexipentiruncicantitruncated 7-cube

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

Alternate names

  • Petiterigreatoprismated hepteract (acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Hexipentistericantitruncated 7-cube

Hexipentistericantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,2,4,5,6{4,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Petitericelligreatorhombihepteract (acronym: putcagroh) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Omnitruncated 7-cube

Omnitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,2,3,4,5,6{3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7,
Properties convex

The omnitruncated 7-cube is the largest uniform 7-polytope in the B7 symmetry of the regular 7-cube. It can also be called the hexipentisteriruncicantitruncated 7-cube which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.

Alternate names

  • Great petated hepteract (Acronym: ) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

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,
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, PhD (1966)
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3o3o3o4x - , x3x3o3o3o3o3x- , x3o3o3x3o3o4x - , x3x3x3o3o3o4x - , x3x3o3x3o3o4x - , x3o3x3x3o3o4x - , x3o3x3o3o3x4x - , x3o3x3o3x3o4x - , x3x3o3o3o3x4x - , x3x3x3x3o3o4x - , x3x3x3o3x3o4x - , x3x3o3x3x3o4x - , x3o3x3x3x3o4x - , x3x3x3oxo3x4x - , x3x3x3x3x3o4x - , x3x3x3o3x3x4x - , x3x3o3x3x3x4x - , x3x3x3x3x3x4x -

External links

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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