Flexing infinite frameworks with applications to braced Penrose tilings

TL;DR

This paper characterizes when infinite planar frameworks are flexible using NAC-colorings, applying the results to Penrose tilings and symmetric frameworks to determine conditions for flexibility and symmetry preservation.

Contribution

It extends flexibility characterization to countably infinite frameworks and applies it to Penrose tilings with braced rhombi and symmetric frameworks.

Findings

A connected graph with a countable vertex set is flexible iff it has a NAC-coloring.

Characterization of flexibility for Penrose rhombus tilings with braced rhombi.

Conditions for flexibility and symmetry preservation in frameworks with rotational symmetry.

Abstract

A planar framework -- a graph together with a map of its vertices to the plane -- is flexible if it allows a continuous deformation preserving the distances between adjacent vertices. Extending a recent previous result, we prove that a connected graph with a countable vertex set can be realized as a flexible framework if and only if it has a so-called NAC-coloring. The tools developed to prove this result are then applied to frameworks where every 4-cycle is a parallelogram, and countably infinite graphs with $n$-fold rotational symmetry. With this, we determine a simple combinatorial characterization that determines whether the 1-skeleton of a Penrose rhombus tiling with a given set of braced rhombi will have a flexible motion, and also whether the motion will preserve 5-fold rotational symmetry.