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Steric 6-cubes

Orthogonal projections in D5 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
Schläfli symbol
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Runcinated demihexeract
  • Runcinated 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

Stericantic 6-cube

Stericantic 6-cube
Type
Schläfli symbol
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Runcitruncated demihexeract
  • Runcitruncated 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

Steriruncic 6-cube

Steriruncic 6-cube
Type
Schläfli symbol
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Runcicantellated demihexeract
  • Runcicantellated 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

Steriruncicantic 6-cube

Steriruncicantic 6-cube
Type
Schläfli symbol
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Runcicantitruncated demihexeract
  • Runcicantitruncated 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

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, wiley.com,
      • (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.
  • x3o3o *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax
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