The spectrum of flexible 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. It was then conjectured that the interior volume of such a flexible polyhedron would remain constant, and this well-known ''Bellows conjecture'' was proved by Sabitov in 2006 using algebraic methods. Recently, V. Alexandrov has conjectured another related but stronger statement: all eigenvalues of the Laplace operator, in a flexible polyhedron with Dirichlet boundary conditions, remain constant. The goal of the project would be to check experimentally this conjecture, by computing numerically the first eigenvalues of the Laplace operator in a few examples of flexible 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