## 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.