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Steric 6-cubes

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6-demicube
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Steric 6-cube
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Stericantic 6-cube
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Steriruncic 6-cube
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Stericruncicantic 6-cube
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Orthogonal projections in D6 Coxeter plane

In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.

Steric 6-cube

Steric 6-cube
Type uniform 6-polytope
Schläfli symbol t0,3{3,3}
h4{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 480
Vertex figure
Coxeter groups D6,
Properties convex

Alternate names

  • Runcinated demihexeract/6-demicube
  • Small prismated hemihexeract (Acronym sophax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a steric 6-cube 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

Related polytopes

Dimensional family of steric n-cubes
n 5 6 7 8

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Steric
figure
Coxeter
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=
Schläfli h4{4,3} h4{4,3} h4{4,3} h4{4,3}

Stericantic 6-cube

Stericantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,3{3,3}
h2,4{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 12960
Vertices 2880
Vertex figure
Coxeter groups D6,
Properties convex

Alternate names

  • Runcitruncated demihexeract/6-demicube
  • Prismatotruncated hemihexeract (Acronym pithax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 2880 vertices of a stericantic 6-cube 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

Steriruncic 6-cube

Steriruncic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,3{3,3}
h3,4{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 7680
Vertices 1920
Vertex figure
Coxeter groups D6,
Properties convex

Alternate names

  • Runcicantellated demihexeract/6-demicube
  • Prismatorhombated hemihexeract (Acronym prohax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 1920 vertices of a steriruncic 6-cube 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

Steriruncicantic 6-cube

Steriruncicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,3{3,3}
h2,3,4{4,3}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 17280
Vertices 5760
Vertex figure
Coxeter groups D6,
Properties convex

Alternate names

  • Runcicantitruncated demihexeract/6-demicube
  • Great prismated hemihexeract (Acronym gophax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 5760 vertices of a steriruncicantic 6-cube 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

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 *b3o3x3o - sophax)
  2. Klitzing, (x3x3o *b3o3x3o - pithax)
  3. Klitzing, (x3o3o *b3x3x3o - prohax)
  4. Klitzing, (x3x3o *b3x3x3o - gophax)

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 *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax

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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