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Pentic 6-cubes

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6-demicube
(half 6-cube)
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Pentic 6-cube
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Penticantic 6-cube
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Pentiruncic 6-cube
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Pentiruncicantic 6-cube
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Pentisteric 6-cube
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Pentistericantic 6-cube
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Pentisteriruncic 6-cube
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Pentisteriruncicantic 6-cube
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Orthogonal projections in D6 Coxeter plane

In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope.

There are 8 pentic forms of the 6-cube.

Pentic 6-cube

Pentic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,4{3,3}
h5{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 1440
Vertices 192
Vertex figure
Coxeter groups D6,
Properties convex

The pentic 6-cube, , has half of the vertices of a pentellated 6-cube, .

Alternate names

  • Stericated 6-demicube/demihexeract
  • Small cellated hemihexeract (Acronym: sochax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry
Coxeter plane D6 D5
Graph
Dihedral symmetry
Coxeter plane D4 D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Penticantic 6-cube

Penticantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,4{3,3}
h2,5{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 9600
Vertices 1920
Vertex figure
Coxeter groups D6,
Properties convex

The penticantic 6-cube, , has half of the vertices of a penticantellated 6-cube, .

Alternate names

  • Steritruncated 6-demicube/demihexeract
  • cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±3,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry
Coxeter plane D6 D5
Graph
Dihedral symmetry
Coxeter plane D4 D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentiruncic 6-cube

Pentiruncic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,4{3,3}
h3,5{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 10560
Vertices 1920
Vertex figure
Coxeter groups D6,
Properties convex

The pentiruncic 6-cube, , has half of the vertices of a pentiruncinated 6-cube (penticantellated 6-orthoplex), .

Alternate names

  • Stericantellated 6-demicube/demihexeract
  • cellirhombated hemihexeract (Acronym: crohax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry
Coxeter plane D6 D5
Graph
Dihedral symmetry
Coxeter plane D4 D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentiruncicantic 6-cube

Pentiruncicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,4{3,3}
h2,3,5{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 20160
Vertices 5760
Vertex figure
Coxeter groups D6,
Properties convex

The pentiruncicantic 6-cube, , has half of the vertices of a pentiruncicantellated 6-cube or (pentiruncicantellated 6-orthoplex),

Alternate names

  • Stericantitruncated demihexeract, stericantitruncated 7-demicube
  • Great cellated hemihexeract (Acronym: cagrohax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry
Coxeter plane D6 D5
Graph
Dihedral symmetry
Coxeter plane D4 D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentisteric 6-cube

Pentisteric 6-cube
Type uniform 6-polytope
Schläfli symbol t0,3,4{3,3}
h4,5{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 5280
Vertices 960
Vertex figure
Coxeter groups D6,
Properties convex

The pentisteric 6-cube, , has half of the vertices of a pentistericated 6-cube (pentitruncated 6-orthoplex),

Alternate names

  • Steriruncinated 6-demicube/demihexeract
  • Small cellipriamated hemihexeract (Acronym: cophix) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry
Coxeter plane D6 D5
Graph
Dihedral symmetry
Coxeter plane D4 D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentistericantic 6-cube

Pentistericantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,3,4{3,3}
h2,4,5{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 23040
Vertices 5760
Vertex figure
Coxeter groups D6,
Properties convex

The pentistericantic 6-cube, , has half of the vertices of a pentistericantellated 6-cube (pentiruncitruncated 6-orthoplex), .

Alternate names

  • Steriruncitruncated demihexeract/7-demicube
  • cellitruncated hemihexeract (Acronym: capthix) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry
Coxeter plane D6 D5
Graph
Dihedral symmetry
Coxeter plane D4 D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentisteriruncic 6-cube

Pentisteriruncic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,3,4{3,3}
h3,4,5{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 15360
Vertices 3840
Vertex figure
Coxeter groups D6,
Properties convex

The pentisteriruncic 6-cube, , has half of the vertices of a pentisteriruncinated 6-cube (penticantitruncated 6-orthoplex), .

Alternate names

  • Steriruncicantellated 6-demicube/demihexeract
  • Celliprismatorhombated hemihexeract (Acronym: caprohax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry
Coxeter plane D6 D5
Graph
Dihedral symmetry
Coxeter plane D4 D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentisteriruncicantic 6-cube

Pentisteriruncicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,3,4{3,3}
h2,3,4,5{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 34560
Vertices 11520
Vertex figure
Coxeter groups D6,
Properties convex

The pentisteriruncicantic 6-cube, , has half of the vertices of a pentisteriruncicantellated 6-cube (pentisteriruncicantitruncated 6-orthoplex), .

Alternate names

  • Steriruncicantitruncated 6-demicube/demihexeract
  • Great cellated hemihexeract (Acronym: gochax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry
Coxeter plane D6 D5
Graph
Dihedral symmetry
Coxeter plane D4 D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Related polytopes

There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

D6 polytopes

h{4,3}
h2{4,3}
h3{4,3}
h4{4,3}
h5{4,3}
h2,3{4,3}
h2,4{4,3}
h2,5{4,3}

h3,4{4,3}
h3,5{4,3}
h4,5{4,3}
h2,3,4{4,3}
h2,3,5{4,3}
h2,4,5{4,3}
h3,4,5{4,3}
h2,3,4,5{4,3}

Notes

  1. Klitzing, (x3o3o *b3o3x3o3o - sochax)
  2. Klitzing, (x3x3o *b3o3x3o3o - cathix)
  3. Klitzing, (x3o3o *b3x3x3o3o - crohax)
  4. Klitzing, (x3x3o *b3x3x3o3o - cagrohax)
  5. Klitzing, (x3o3o *b3o3x3x3x - cophix)
  6. Klitzing, (x3x3o *b3o3x3x3x - capthix)
  7. Klitzing, (x3o3o *b3x3x3x3x - caprohax)
  8. Klitzing, (x3x3o *b3x3x3x3o - gochax)

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, Ph.D.
  • Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o3o *b3o3x3o3o - sochax, x3x3o *b3o3x3o3o - cathix, x3o3o *b3x3x3o3o - crohax, x3x3o *b3x3x3o3o - cagrohax, x3o3o *b3o3x3x3x - cophix, x3x3o *b3o3x3x3x - capthix, x3o3o *b3x3x3x3x - caprohax, x3x3o *b3x3x3x3o - gochax

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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