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

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.

• Rhombic dodecahedron? – Ilya Bogdanov Sep 16 '19 at 20:27
• @IlyaBogdanov You are absolutely right. And also the rhombic triacontahedron. I have overlooked these. So I guess my question now is whether there are any more? – M. Winter Sep 16 '19 at 20:31
• What about Cartesian products of $d$-cubes, rhombic dodecahedrons and rhombic triacontahedrons? – LeechLattice Sep 16 '19 at 23:30
• @Bullet51 I think the cartesian product of a $d_1$-cube and a $d_2$-cube is a $(d_1+d_2)$-cube. You are right about the latter two examples. I already mentioned these in my comment, but I should edit them into my post. – M. Winter Sep 17 '19 at 5:13