# Model Sets: A Survey

@article{Moody2000ModelSA, title={Model Sets: A Survey}, author={Robert V. Moody}, journal={arXiv: Metric Geometry}, year={2000}, pages={145-166} }

Even when reduced to its simplest form, namely that of point sets in euclidean space, the phenomenon of genuine quasi-periodicity appears extraordinary. Although it seems unfruitful to try and define the concept precisely, the following properties may be considered as representative:
discreteness;
extensiveness;
finiteness of local complexity;
repetitivity;
diffractivity;
aperiodicity;
existence of exotic symmetry (optional).

## 127 Citations

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The geometric properties of almost periodicity of model sets (or cut-and-project sets, defined under the weakest hypotheses) are investigated and it is proved that they are almost periodic patterns and thus possess a uniform density.

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A probabilistic construction of model sets arising from cut and project schemes in Euclidean spaces whose associated Delone dynamical systems have positive toplogical entropy.

## References

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It is shown that taking the site occupation of a model set stochastically results, with probabilistic certainty, in well-defined diffractive properties augmented by a constant diffuse background.

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Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal…

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The torus parametrization of quasiperiodic local isomorphism classes is introduced and used to determine the number of elements in such a class with special symmetries or inflation properties. The…

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This contribution deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical…

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### Self-Similarities and Invariant Densities for Model Sets

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Model sets (also called cut and project sets) are generalizations of lattices. Here we show how the self-similarities of model sets are a natural replacement for the group of translations of a…

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This paper gives a rigorous treatment of some aspects of diffraction by aperiodic structures such as quasicrystals. It analyses diffraction in the limit of the infinite system, through an…

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