Flexibility of Penrose frameworks

Abstract

A framework, which is a (possibly infinite) graph together with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the distances between adjacent vertices. The existence of a flexible framework for a given graph is characterised by the existence of a so called NAC-coloring — a surjective edge coloring by red and blue such that each cycle is either monochromatic, or contains at least two red and two blue edges. In this talk, we focus on infinite frameworks obtained as 1-skeleta of parallelogram tilings. We brace some of the parallelograms, namely, they are not allowed to change their shape during a flex. We show that such a structure is flexible if and only if the graph admits a special type of NAC-coloring, called cartesian. Moreover, if this framework is n-fold rotationally symmetric, we can again decide its flexibility by the existence a cartesian NAC-coloring invariant under the symmetry. In particular, we can apply these results to frameworks obtained from (5-fold symmetric) Penrose tilings.