Construction 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. There are a number of known examples of flexible polyhedra. The goal of the project will be to find explicit and exact descriptions of some of those examples, that is, compute the coordinate of the vertices as a function of the deformation parameter. A possible second stage of the project will be to check whether some quantities are conserved in a deformation. It is known that the volume of flexible polyhedra remains constant, as well as other quantities like the total mean curvature. However other quantities might be constant in a flex.

Schedule: to be determined.

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

Difficulty level: Introductory.

Tools: programs in python/sage.

Expected results: Construct explicitly some examples of flexible polyhedra. Those examples might be used later in other EML projects.

FSCT -- University of Luxembourg