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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.

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  • $\begingroup$ Rhombic dodecahedron? $\endgroup$ – Ilya Bogdanov Sep 16 '19 at 20:27
  • $\begingroup$ @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? $\endgroup$ – M. Winter Sep 16 '19 at 20:31
  • $\begingroup$ What about Cartesian products of $d$-cubes, rhombic dodecahedrons and rhombic triacontahedrons? $\endgroup$ – LeechLattice Sep 16 '19 at 23:30
  • $\begingroup$ @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. $\endgroup$ – M. Winter Sep 17 '19 at 5:13
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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.

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