Aperiodic Order: Volume 1, A Mathematical Invitation

A comprehensive introductory monograph on the theory of aperiodic order, with numerous illustrations and examples.

Michael Baake (Author), Uwe Grimm (Author)

9780521869911, Cambridge University Press

Hardback, published 22 August 2013

552 pages, 145 b/w illus. 18 colour illus. 10 tables
24.2 x 16.1 x 3.3 cm, 1.08 kg

'Aperiodic Order is a comprehensive introduction to this relatively new and multidisciplinary field. Sparked by Dan Shechtman's discovery of quasicrystals in 1982, which earned him the 2011 Nobel Prize in Chemistry, the field incorporates crystallography, discrete geometry, dynamical systems, harmonic analysis, mathematical diffraction theory, and more. Because the field spans such disparate fields, advances by one group often go unnoticed by the other. An important goal of this book is to remedy this by unifying and contextualizing results and providing a common language for researchers. … Readers who want to follow up on any details can certainly find a reference in the nearly 30 pages of bibliographic entries. Full of examples, construction techniques, and an array of analytic tools, this book is an outstanding resource for those hoping to enter the field, yet also contains plenty of useful information for seasoned experts.' Natalie Priebe Frank, Mathematical Association of America

Quasicrystals are non-periodic solids that were discovered in 1982 by Dan Shechtman, Nobel Prize Laureate in Chemistry 2011. The underlying mathematics, known as the theory of aperiodic order, is the subject of this comprehensive multi-volume series. This first volume provides a graduate-level introduction to the many facets of this relatively new area of mathematics. Special attention is given to methods from algebra, discrete geometry and harmonic analysis, while the main focus is on topics motivated by physics and crystallography. In particular, the authors provide a systematic exposition of the mathematical theory of kinematic diffraction. Numerous illustrations and worked-out examples help the reader to bridge the gap between theory and application. The authors also point to more advanced topics to show how the theory interacts with other areas of pure and applied mathematics.

Foreword Roger Penrose
Preface
1. Introduction
2. Preliminaries
3. Lattices and crystals
4. Symbolic substitutions and inflations
5. Patterns and tilings
6. Inflation tilings
7. Projection method and model sets
8. Fourier analysis and measures
9. Diffraction
10. Beyond model sets
11. Random structures
A. The icosahedral group
Appendix B. The dynamical spectrum
References
Index.

Subject Areas: Maths for engineers [TBJ], Mathematical physics [PHU], Algebraic geometry [PBMW], Geometry [PBM], Calculus & mathematical analysis [PBK], Algebra [PBF]