Inflating the volume of polyhedra

Abstract

Motivated by physicists wrapping oil drops in ultrathin plastic sheets, we are inter ested in volume-increasing deformations of polyhedra. We implement a constructive proof by Igor Pak in Mathematica. We take as input a tetrahedron and output a submetric deformation that has greater volume. This implies that there is an isom etry that also increases volume. We extend the algorithm to work on any “simple” polyhedron: one all of whose vertices have degree 3. We investigate the relationship between volume and surface area for several of these deformations and discuss our findings