Flexible polyhedral nets in isotropic geometry

Abstract

We study flexible polyhedral nets in isotropic geometry. This geometry has a degenerate metric, but there is a natural notion of flexibility. We study infinitesimal and finite flexibility, and classify all finitely flexible polyhedral nets of arbitrary size. We show that there are just two classes, in contrast to Izmestiev's rather involved classification in Euclidean geometry, for size 3x3 only. Using these nets to initialize the optimization algorithms, we turn them into approximate Euclidean mechanisms. We also explore the smooth versions of these classes.