Linear zero mode spectra for quasicrystals

TL;DR

This paper introduces the concept of linear zero mode spectra for aperiodic frameworks like quasicrystals, linking local flexes to spectral properties and providing geometric characterizations.

Contribution

It defines new spectra for quasicrystals based on local flexes and establishes their geometric determination, extending spectral analysis beyond periodic structures.

Findings

The spectra coincide for certain quasicrystal frameworks.

Spectra are determined by the geometry of the frameworks.

Connections between local flexes and spectral properties are established.

Abstract

A converse is given to the well-known fact that a hyperplane localised zero mode of a crystallographic bar-joint framework gives rise to a line or lines in the zero mode (RUM) spectrum. These connections motivate definitions of linear zero mode spectra, for an aperiodic bar-joint framework $G$, that are based on relatively dense sets of linearly localised flexes. For a Delone framework in the plane the limit spectrum ${\bf L}_{lim}(G, a)$ is defined in this way, as a subset of the reciprocal space for a reference basis $a$ of the ambient space. A smaller spectrum, the slippage spectrum ${\bf L}_{slip}(G, a)$, is also defined. In the case of the quasicrystal parallelogram frameworks associated with regular multi-grids, in the sense of de Bruijn and Beenker, these spectra coincide and are determined in terms of the geometry of $G$.