Let (P) be a d-dimensional polytope with d>1. (Q1) Is there any procedure to determine all symmetries of (P) (for example all symmetries with respect to hyperplanes)? (Q2) How can we obtain a dissection of (P) into new polytopes (P1), (P2), ..., (Pm) in such a way that the maximum number of vertices on (P1), (P2), ..., (Pm) is minimal?
I'm trying to understand your question better. How does the polytope whose symmetries you wish to understand arise? Are its vertices or bounding hyperplanes obtained from a random process? Is it V or H presented? How much of its structure do you have available? For example, do you have its face lattice?
I haven't seen a truly generic method for doing this, and in most contexts I'm familiar with where symmetries are being investigated the goal has been either to establish existence (i.e., given a symmetry group is there a polytope of a particular class with that symmetry type) or address other combinatorial questions via the symmetry group.
Your second question is a bit further outside my experience area, but I think it would help to have a better idea what constraints you have on the dissection, e.g., are you allowed to introduce new vertices on the boundary or interior of the polytope to achieve the dissection?
Thank you very much for answering.
About the first question. I deal with parameterized polytopes in both V or H presentation ( since I can switch from one to another). My aim is to compute the exact d-volume of each polytope. Although I am able to compute such volume using several well known algorithms, symmetries are very important for me.
About the second question. Still in volume computations, I would like to triangulate polytopes in terms a minimum number of simplices. Any suggestion on how to achieve such triangulations is welcome.
- Badges
- Science topic