Parallelogram frameworks and flexible quasicrystals

TL;DR

This paper characterizes the flexibility and rigidity of parallelogram frameworks and quasicrystal structures, providing explicit bases for their flex spaces and analyzing their zero mode spectra.

Contribution

It introduces explicit bases for the flex spaces of parallelogram frameworks and applies these results to quasicrystal frameworks, identifying conditions for rigidity and flexibility.

Findings

Explicit free basis for the flex space of parallelogram frameworks

Characterization of rigid bracing patterns in parallelogram frameworks

Identification of finite-dimensional flex spaces in quasicrystal frameworks

Abstract

The first-order flex space of the bar-joint framework $G_P$ of a parallelogram tiling $P$ is determined in terms of an explicit free basis. Applications are given to braced parallelogram frameworks and to quasicrystal frameworks associated with multigrids in the sense of de Bruijn and Beenker. In particular we characterise rigid bracing patterns, identify quasicrystal frameworks with finite dimensional flex spaces, and define a zero mode spectrum.