Runcic 6-cubes

In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

Runcic 6-cube

The Cartesian coordinates for the vertices of a runcic 6-cube centered at the origin are coordinate permutations:

(ÃÂ±1,ÃÂ±1,ÃÂ±1,ÃÂ±3,ÃÂ±3,ÃÂ±3)

with an odd number of plus signs.

The Cartesian coordinates for the vertices of a runcicantic 6-cube centered at the origin are coordinate permutations:

(ÃÂ±1,ÃÂ±1,ÃÂ±3,ÃÂ±5,ÃÂ±5,ÃÂ±5)

with an odd number of plus signs.

This polytope is based on the 6-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 47 uniform polytopes with D<sub>6</sub> symmetry, 31 are shared by the B<sub>6</sub> symmetry, and 16 are unique:

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 IviÃÂ Weiss, Wiley-Interscience Publication, 1995, wiley.com,
(Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380Ã¢ÂÂ407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559Ã¢ÂÂ591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3Ã¢ÂÂ45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
x3o3o *b3x3o3o, x3x3o *b3x3o3o