Model Sets: A Survey

  title={Model Sets: A Survey},
  author={Robert V. Moody},
  journal={arXiv: Metric Geometry},
  • R. Moody
  • Published 2 February 2000
  • Mathematics
  • arXiv: Metric Geometry
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). 

Diffraction of stochastic point sets: Exactly solvable examples

Stochastic point sets are considered that display a diffraction spectrum of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality.

Spectral notions of aperiodic order

This article uses Delone sets in Euclidean space as the main object class, and gives generalisations in the form of further examples and remarks about the relations between them.

Diffraction of Stochastic Point Sets: Explicitly Computable Examples

Stochastic point processes relevant to the theory of long-range aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases

A Note on Shelling

Several aspects of central versus averaged shelling are summarized, the difference with explicit examples and the obstacles that emerge with aperiodic order are discussed.

On weak model sets of extremal density

Ellis enveloping semigroup for almost canonical model sets of an Euclidean space

We consider certain point patterns of an Euclidean space and calculate the Ellis enveloping semigroup of their associated dynamical systems. The algebraic structure and the topology of the Ellis

A guide to lifting aperiodic structures

Abstract The embedding of a given point set with non-crystallographic symmetry into higher-dimensional space is reviewed, with special emphasis on the Minkowski embedding known from number theory.

Pure Point Diffraction Implies Zero Entropy for Delone Sets with Uniform Cluster Frequencies

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy zero if it has uniform cluster frequencies and is pure

Model sets, almost periodic patterns, uniform density and linear maps

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.

Model sets with positive entropy in Euclidean cut and project schemes

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



Diffractive point sets with entropy

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.


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

The torus parametrization of quasiperiodic LI-classes

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

A Guide to Mathematical Quasicrystals

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

Diffraction from visible lattice points and kth power free integers

Self-Similarities and Invariant Densities for Model Sets

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

Dynamics of self-similar tilings

  • B. Solomyak
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1997
This paper investigates dynamical systems arising from the action by translations on the orbit closures of self-similar and self-affine tilings of ${\Bbb R}^d$. The main focus is on spectral

Space tilings and local isomorphism

We prove for a large class of tilings that, given a finite tile set, if it is possible to tile Euclideann-space with isometric copies of this set, then there is a tiling with the ‘local isomorphism

On diffraction by aperiodic structures

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

Sphere Packings, Lattices and Groups

The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to