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Runcic 5-cubes

In five-dimensional geometry, a runcic 5-cube, runcic 5-demicube or 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

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

Related polytopes

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

Runcicantic 5-cube

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

Related polytopes

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

Related polytopes

These polytopes are based on the 5-demicube, a member of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 uniform 5-polytopes that can be constructed from the D<sub>5</sub> symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.

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 Ivić 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

External links