Cantic 6-cube explained

bgcolor=#e7dcc3 colspan=2	Cantic 6-cube Truncated 6-demicube
bgcolor=#ffffff align=center colspan=2	D6 Coxeter plane projection
Type	uniform polypeton
Schläfli symbol	t_0,1 h₂
Coxeter-Dynkin diagram	=
5-faces	76
4-faces	636
Cells	2080
Faces	3200
Edges	2160
Vertices	480
Vertex figure	v[{ }x{3,3}]
Coxeter groups	D₆, [3<sup>3,1,1</sup>]
Properties	convex

In six-dimensional geometry, a cantic 6-cube (or a truncated 6-demicube) is a uniform 6-polytope.

Alternate names

The Cartesian coordinates for the 480 vertices of a cantic 6-cube centered at the origin and edge length 6 are coordinate permutations:

(±1,±1,±3,±3,±3,±3)with an odd number of plus signs.

There are 47 uniform polytopes with D₆ symmetry, 31 are shared by the B₆ 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 Ivic Weiss, Wiley-Interscience Publication, 1995, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  - (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.
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