Are there any more polytopes whose 2-faces are identical 4-gons?

Question

What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds

• $P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
• all 2-faces of $P$ are 4-gons (not necessarily squares, or rectangles).

I know the $d$-cubes, rhombic dodecahedron and rhombic triacontahedron. Are there any others?

I expect other such a polytope, if at all, then only in $d\ge 4$. Maybe a slight modification of a neighborly cubical polytope as constructed by Ziegler here, but as they are, they have two 2-face orbits.

Answer 1

In fact, there are many to be found on Wikipedia under isogonal figures, even in three dimensions.

Examples in dimension four are obtained as dual polytope of runcinated 4-simplex or runcinated 24-cell. This works, because these both runcinations are

1. edge-transitive, and
2. each edge is contained in exactly four facets.

Since in the dual edges become 2-faces and facets become vertices, the resulting dual polytopes will be 2-face-transitive, and each 2-face will have four vertices, that is, they will be 4-gons.

If I am not mistaken, the cells of these two examples are trigonal trapezohedron (aka. elongated cubes), and these are 3-dimensional examples also found in the Wikipedia list mentioned above.