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A converse is given to the well-known fact that a hyperplane localised zeromode of a crystallographic bar-joint framework gives rise to a line or lines inthe zero mode (RUM) spectrum. These connections motivate definitions of linearzero mode spectra, for an aperiodic bar-joint framework $G$, that are based onrelatively dense sets of linearly localised flexes. For a Delone framework inthe plane the limit spectrum ${\bf L}_{lim}(G, a)$ is defined in this way, as asubset 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 alsodefined. In the case of the quasicrystal parallelogram frameworks associatedwith regular multi-grids, in the sense of de Bruijn and Beenker, these spectracoincide and are determined in terms of the geometry of $G$.

Linear zero mode spectra for quasicrystals