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Runcic 5-cubes

Concept in geometry

Concept in geometry

Orthogonal projections in B5 Coxeter plane

In six-dimensional geometry, a runcic 5-cube or (runcic 5-demicube, runcihalf 5-cube) is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

Runcic 5-cube

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

Alternate names

  • Cantellated 5-demicube/demipenteract
  • Small rhombated hemipenteract (sirhin) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 960 vertices of a runcic 5-cubes centered at the origin are coordinate permutations: : (±1,±1,±1,±3,±3) with an odd number of plus signs.

Images

It has half the vertices of the runcinated 5-cube, as compared here in the B5 Coxeter plane projections:

[[File:5-demicube t02 B5.svg240px]]Runcic 5-cube[[File:5-cube t03.svg240px]]Runcinated 5-cube

Runcicantic 5-cube

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

Alternate names

  • Cantitruncated 5-demicube/demipenteract
  • Great rhombated hemipenteract (girhin) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a runcicantic 5-cube centered at the origin are coordinate permutations: : (±1,±1,±3,±5,±5) with an odd number of plus signs.

Images

It has half the vertices of the runcicantellated 5-cube, as compared here in the B5 Coxeter plane projections:

[[File:5-demicube t012 B5.svg240px]]Runcicantic 5-cube[[File:5-cube t023.svg240px]]Runcicantellated 5-cube

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 *b3x3o - sirhin, x3x3o *b3x3o - girhin
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