Orthogonal projections
Posted: 22 Mar 2014, 19:53
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 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
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.
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
Code: Select all
$RTP = projection($RT,[1,2]); I built an arbitrary rotation matrix $m and did
Code: Select all
$rotRT = transform($RT,$m);
$RTP = projection($rotRT,[2,3]);
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.