On the realization space of the cube

We consider the realization space of the \(d\)-dimensional cube, and show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. We use this fact to define an analog of the connected sum construction for cubical \(d\)-polytopes, and apply this construction to certain cubical \(d\)-polytopes to conclude that the rays spanned by \(f\)-vectors of cubical \(d\)-polytopes are dense in Adin's cone. The connectivity result on cubes extends to any product of simplices, and further, it shows the respective realization spaces are contractible.