Computing a matrix to understand the rigidity of polyhedra

Goal: It has been known since Legendre and Cauchy that convex polyhedra in Euclidean space are always rigid, that is, they cannot be deformed without changing the ''shape'' or their faces. However, non-convex polyhedra can be flexible, the first examples were discovered by R. Connelly in 1968. The motivation for this project is a conjecture: any non-convex polyhedron with vertices in convex position which is decomposable (can be cut into convex pieces without creating any new vertex) is infinitesimally rigid. One possible approach to this conjecture is to search for counter-examples. The project targets the first step in this search: the computation, for any polyhedron equipped with a decomposition of its interior into simplices, of a symmetric matrix which has non-zero kernel if and only if the polyhedron is infinitesimally flexible. This computation is elementary but somewhat involved, and the project will start by computing the dihedral angles of a simplex as a function of its edge lengths, and their derivatives. In case the project advances well, it would be possible to go on to the next steps of the search for counter-examples to the infinitesimal rigidity conjecture for non-convex polyhedra.

Schedule: to be determined.

Persons involved: Advised by Jean-Marc Schlenker, NN.

Difficulty level: medium.

Tools: programs in python/sage, use of computational tools to compute the first eigenvalues of the Laplace operator.

Expected results: either disprove the conjecture, or give some evidence that it might be correct by showing that, up to some approximation, the first few eigenvalues of the Laplace operator remain constant in an isometric deformation in a few simple examples.

FSCT -- University of Luxembourg