Parallelogram frameworks and flexible quasicrystals

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.

Bracing frameworks consisting of parallelograms

“…The equivalence of (i) and (iv) was proven for the finite case in [11] and for the infinite case in [7]. For P-frameworks obtained from parallelogram tilings the equivalence with (ii) follows from [20]. Since the number of connected components of the bracing graph as defined in [11] is the same as the number of angle-preserving classes, Theorem 1.1 of [11] is implied by Corollary 1.18.…”

“…Such frameworks are for instance interesting in physics due to its relation with quasicrystals [24]. Rhombic tilings and their flexibility were found to be interesting for arts [23] and have later been formalized [8,11,20]. Again the connectivity of an auxiliary graph plays an important role.…”

Linear zero mode spectra for quasicrystals

“…More generally, in our companion paper [20] we have considered parallelogram frameworks G P for parallelogram tilings P of the plane which are determined by a regular multigrid in the sense of De Bruijn [6] and Beenker [4]. Once again, ribbons are linearly localised although, for reasons of asymmetry, their directions, which are taken to define the ribbon figure RF (P ), need not coincide with the perpendicular line directions.…”

“…Example 4.12. For a Penrose rhomb tiling P for a regular pentagrid [6], [20] Consider a rational approximation to a Penrose rhomb tiling P by a periodic parallelogram tiling P ′ . As is well known, one can construct such approximants by the projection method [1], [9], [10], [23].…”

Linear zero mode spectra for quasicrystals

“…For these identifications we use the characterisation of infinitesimal flexes of parallelogram frameworks obtained in our companion article [20], where we also give an explicit formula for RF (P ) in terms of the tile geometry of P .…”