Orthogonal projections

[kundor]

Say a projection of a polytope P onto a hyperplane is regular if the normal vector to the hyperplane is not parallel to any proper face of P.
The number of combinatorial types of regular projections of a polytope is finite.
I want to find m_j(P), the maximum number of j-faces of a regular projection of P.

Can I do this with polymake? For instance, I am examining the rhombic triacontahedron. By using

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$RTP = projection($RT,[1,2]);

I can project it to coordinate hyperplanes, where it always comes out as an octagon. But just examining the default visualization from $RT->VISUAL(); shows a projection as a dodecagon.
I built an arbitrary rotation matrix $m and did

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$rotRT = transform($RT,$m); $RTP = projection($rotRT,[2,3]);

and got a decagon. Using a different rotation matrix netted me a 14-gon.

How can I know if I've gotten the maximum? Is there a choice of rotations which will give me all the combinatorial types of regular projections? Any hints would be appreciated.

[joswig]

It seems like the various (affine) projection directions define a complete fan structure, that I never thought about before. Therefore, unfortunately, there is no standard function in polymake to help you out. Sorry.

Looks interesting though. Keep me posted if you have results.