Pentic 6-cubes


6-demicube
(half 6-cube)
=
Pentic 6-cube
=
Penticantic 6-cube
=

Pentiruncic 6-cube
=
Pentiruncicantic 6-cube
=
Pentisteric 6-cube
=

Pentistericantic 6-cube
=
Pentisteriruncic 6-cube
=
Pentisteriruncicantic 6-cube
=
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{3,34,1}
h5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 1440
Vertices 192
Vertex figure
Coxeter groups D6, [33,1,1]
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)[1]

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 [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Penticantic 6-cube

Penticantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,4{3,34,1}
h2,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 9600
Vertices 1920
Vertex figure
Coxeter groups D6, [33,1,1]
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)[2]

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 [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentiruncic 6-cube

Pentiruncic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,4{3,34,1}
h3,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 10560
Vertices 1920
Vertex figure
Coxeter groups D6, [33,1,1]
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)[3]

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 [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentiruncicantic 6-cube

Pentiruncicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,4{3,32,1}
h2,3,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 20160
Vertices 5760
Vertex figure
Coxeter groups D6, [33,1,1]
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)[4]

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 [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentisteric 6-cube

Pentisteric 6-cube
Type uniform 6-polytope
Schläfli symbol t0,3,4{3,34,1}
h4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 5280
Vertices 960
Vertex figure
Coxeter groups D6, [33,1,1]
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)[5]

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 [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentistericantic 6-cube

Pentistericantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,3,4{3,34,1}
h2,4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 23040
Vertices 5760
Vertex figure
Coxeter groups D6, [33,1,1]
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)[6]

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 [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentisteriruncic 6-cube

Pentisteriruncic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,3,4{3,34,1}
h3,4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 15360
Vertices 3840
Vertex figure
Coxeter groups D6, [33,1,1]
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)[7]

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 [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentisteriruncicantic 6-cube

Pentisteriruncicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,3,4{3,32,1}
h2,3,4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 34560
Vertices 11520
Vertex figure
Coxeter groups D6, [33,1,1]
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)[8]

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 [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

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

D6 polytopes

h{4,34}
h2{4,34}
h3{4,34}
h4{4,34}
h5{4,34}
h2,3{4,34}
h2,4{4,34}
h2,5{4,34}

h3,4{4,34}
h3,5{4,34}
h4,5{4,34}
h2,3,4{4,34}
h2,3,5{4,34}
h2,4,5{4,34}
h3,4,5{4,34}
h2,3,4,5{4,34}

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 [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, 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
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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