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arXiv · 2503.22457

The Rigid Unit Mode spectrum for symmetric frameworks

Eleftherios Kastis·Derek Kitson
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Abstract

We establish several fundamental properties of the Rigid Unit Mode (RUM) spectrum for symmetric frameworks with a discrete abelian symmetry group and arbitrary linear constraints. In particular, we identify a nonempty subset of the RUM spectrum which derives from the joint eigenvalues of generators for the linear part of the symmetry group. These joint eigenvalues give rise to $\chi$-symmetric flexes which span the space of translations for the framework. We show that the RUM spectrum is a union of Bohr-Fourier spectra arising from twisted almost-periodic flexes of the framework. We also characterise frameworks for which every almost periodic flex is a translation.

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Eleftherios Kastis, Derek Kitson. 2025-03-28. The Rigid Unit Mode spectrum for symmetric frameworks. https://arxiv.org/abs/2503.22457

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