Cubical polytope

A polytope is cubical if all its facets are combinatorial hypercubes.

The existence of a cubical d-polytope with an odd number of facets is a nontrivial problem. For d = 3 this is easy. The d = 4 case was positively resolved in 2012 by Schwartz and Ziegler using a 16,533-cell construction based on a prism of Boy's surface. No such polytope exists for d = 6, 8, 9, or 10, while 5 and 7 remain open.