Book Chapter10.1007/978-3-662-04253-3_6
Model Sets: A Survey
TL;DR: Even when reduced to its simplest form, namely that of point sets in euclidean space, the phenomenon of genuine quasi-periodicity appears extraordinary as discussed by the authors Although it seems unfruitful to try and define the concept precisely, the following properties may be considered as representative.
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Abstract: 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)
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Citations
Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra
Michael Baake,Daniel Lenz +1 more
TL;DR: In this paper, the authors consider topological dynamical systems arising from locally compact Abelian groups on compact spaces of translation bounded measures and show that such a system has a pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
174
Characterization of model sets by dynamical systems
TL;DR: In this paper, it is shown how regular model sets can be characterized in terms of the regularity properties of their associated dynamical systems, and that regular models are as close to periodic sets as possible among repetitive aperiodic sets.
138
Characterizations of model sets by dynamical systems
TL;DR: In this article, it is shown how regular model sets can be characterized in terms of regularity properties of their associated dynamical systems, and that regular models are as close to periodic sets as possible among repetitive aperiodic sets.
137
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Weighted Dirac combs with pure point diffraction
Michael Baake,Robert V. Moody +1 more
TL;DR: In this article, a class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated, and a sufficient set of conditions to ensure that the diffraction measure is a pure point measure is presented.
121
What is Aperiodic Order
TL;DR: The theory of aperiodic order is concerned with the development of ideas stimulated by the discovery of quasicrystals as mentioned in this paper, and it has been studied in a more general audience.
References
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Sphere packings, lattices, and groups
John H. Conway,Neil J. A. Sloane,Eiichi Bannai +2 more
- 01 Dec 1987
TL;DR: 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?
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Recurrence in Ergodic Theory and Combinatorial Number Theory
Harry Furstenberg
- 14 Jul 2014
TL;DR: The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press, to preserve the original texts of these important books while presenting them in durable paperback editions.
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Über die Gleichverteilung von Zahlen mod. Eins
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Undecidability and Nonperiodicity for Tilings of the Plane.
TL;DR: In this article, the authors considered the problem of finding a set of tiles that can be used to tile the plane by translation, assuming that the tiles are to be placed with their vertices at lattice points and abutting edges have the same color.
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Quasicrystals and geometry
Marjorie Senechal
- 01 Jan 1995
TL;DR: An atlas of tiling transforms is given in this article, along with a mathematical toolbag for tiling transform analysis and a diagram of the Penrose tilings of the plane.