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The (Grünbaum, B., 2010) reference in What are the known convex polyhedra with congruent faces? establishes that for $n=1$, the answer is 30, i.e. the convex monohedral rhombic polyhedron with the greatest number of faces is the rhombic triacontahedron (dual of the icosidodecahedron), which happens also to be isohedral (but I wasn't insisting on that).

For $n=2$ there is a well-known https://en.wikipedia.org/wiki/Rhombic_enneacontahedron with 90 faces, but is that the maximum number of faces possible with just two distinct rhombuses? And if this happens to be the maximum, then again we have that there is just one orbit of each type of rhombus under the symmetries of the polyhedron, even though that wasn't required.

For $n=3$ I'm aware of a 132-face rhombic polyhedron: https://geometryka.wordpress.com/2016/11/23/kubus-rhombendodekaeder/#more-756 but I'd be slightly surprised if this is the maximum, simply because adding a second type of face allowed the number of faces to grow by a factor of three and 132 is a much smaller multiple of 90. (Admittedly, that's a weak heuristic.) This polyhedron, however, has two orbits of the middle-eccentricity rhombus; can one do as well with just one orbit of each type? (Seems doubtful.) And this polyhedron is somehow based on a subdivision of the rhombic dodecahedron; could we do something analogous with the rhombic triacontahedron to get more faces with only three rhombus types?

For larger $n$, I have no information but presumably the maximum strictly increases with $n$. I have no sense of how fast it might grow, however. And presumably the maximum will have multiple orbits of some of the types of rhombuses, but I don't know that for sure, either.

Glen Whitney
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  • Hmm, assuming there are no rhombic polyhedra with an axis of rotation of greater symmetry than fivefold, the maximum size of a non-axial point group in $\Bbb{R}^3$ will limit the number of faces with $n$ types and only one orbit of each type to $120n$ (which is not achieved by any of the above examples anyway, nor does any individual orbit of faces have 120 members). – Glen Whitney Oct 03 '22 at 17:41
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    Ahh, actually polar zonohedra can have $D_n$ symmetry for any $n$. But for example, the 10-zonohedron has five rhombus types in general (perhaps fewer in some particular proportions) and 90 faces. So it seems unlikely these will be the record holders for small $n$. – Glen Whitney Oct 04 '22 at 16:28
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    Apparently the max for $n=15$ is at least 870... – Glen Whitney Oct 04 '22 at 16:45
  • Each edge is incident with $2$ faces, and each face is incident with $4$ edges, so the number of edge-face pairs is $2E=4F$. Applying Euler's formula: $$2=V-E+F=V-2F+F=V-F$$ So the maximum number of faces is directly related to the maximum number of vertices. I don't know if that helps. – mr_e_man Nov 08 '22 at 16:12

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