Symbol functions for symmetric frameworks

Kastis, Eleftherios; Kitson, Derek; McCarthy, John E.

doi:10.1016/j.jmaa.2020.124895

Cited by 8 publications

(10 citation statements)

References 15 publications

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“…More results from an operator-theoretic perspective were initiated in [28], which gave applications on factor periodic flexes and the rigid unit mode (RUM) spectrum were initiated. Further studies on the RUM spectrum are discussed in [21].…”

Section: Periodic Frameworkmentioning

confidence: 99%

“…This has motivated a number of groups to explore symmetric rigidity over the past two decades. We refer the reader to [5,11,12,15,17,20,21,25,27,31] and the references therein for details.…”

Section: Introductionmentioning

confidence: 99%

“…As in the generic case, for symmetrically generic frameworks, this is equivalent to having a symmetry preserving infinitesimal flex, which can be expressed as a matrix condition using an analogue of the well-known rigidity matrix (details on this may be found in [31]). Complete combinatorial descriptions of symmetric, but otherwise generic, rigidity for various symmetry groups, including all cyclic groups, have been obtained in the plane [20,21,25].…”

Section: Introductionmentioning

confidence: 99%

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Quotient graphs of symmetrically rigid frameworks

Dewar¹,

Grasegger²,

Kastis³

et al. 2022

Preprint

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A natural problem in combinatorial rigidity theory concerns the determination of the rigidity or flexibility of bar-joint frameworks in R d that admit some non-trivial symmetry. When d = 2 there is a large literature on this topic. In particular, it is typical to quotient the symmetric graph by the group and analyse the rigidity of symmetric, but otherwise generic frameworks, using the combinatorial structure of the appropriate group-labelled quotient graph. However, mirroring the situation for generic rigidity, little is known combinatorially when d ≥ 3. Nevertheless in the periodic case, a key result of Borcea and Streinu [7] characterises when a quotient graph can be lifted to a rigid periodic framework in R d . We develop an analogous theory for symmetric frameworks in R d . The results obtained apply to all finite and infinite 2-dimensional point groups, and then in arbitrary dimension they concern a wide range of infinite point groups, sufficiently large finite groups and groups containing translations and rotations. For the case of finite groups we also derive results concerning the probability of assigning group labels to a quotient graph so that the resulting lift is symmetrically rigid in R d .

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Section: Periodic Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quotient graphs of symmetrically rigid frameworks

Dewar¹,

Grasegger²,

Kastis³

et al. 2022

Preprint

Self Cite

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“…The matrix function is known as the symbol function of the crystal framework with respect to the periodicity basis a. See [18], [19] and [15] for further details.…”

Section: Zero Mode Spectra For Quasicrystalsmentioning

confidence: 99%

Parallelogram frameworks and flexible quasicrystals

Power¹

2021

Mathematical Proceedings of the Royal Irish Academy

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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.

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“…The matrix function is known as the symbol function of the crystal framework with respect to the periodicity basis a. See [18], [19] and [15] for further details. 4.2.…”

Section: Zero Mode Spectra For Quasicrystalsmentioning

confidence: 99%