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Aperiodic Order, Volume 1: A Mathematical Invitation

Michael Baake and Uwe Grimm
Cambridge University Press
Publication Date: 
Number of Pages: 
Encyclopedia of Mathematics and Its Applications 149
[Reviewed by
Natalie Priebe Frank
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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.

The foreword by Roger Penrose gives some history of the field from a surprisingly personal vantage point, providing anecdotes involving M. C. Escher and Dan Shechtman. Penrose describes the origins of his own interest in tilings and the influences that were part of his discovery of what would become the famous “Penrose tilings.” He also likens this book in importance to the field to Branko Grünbaum and G. C. Shephard’s Tilings and PatternsPenrose’s foreword and the introduction by the authors provide a rich historical context that motivates the study of aperiodic order.

The main objects in Aperiodic Order are those that can model atomic structure: tilings and point sets in Euclidean space and sequences in one and many dimensions. Necessary background such as lattices, Delone sets, Voronoi tilings, and the algebraic structures that help analyze them are discussed. Fundamental ways to quantify properties of those objects via metrics and symmetries are also introduced in a way that is useful for the aperiodic situation.

A series of chapters expose the two most-studied ways aperiodically ordered objects are constructed, broadly termed the projection and substitution methods. The details and variations of these methods in both the symbolic (sequences) and geometric (tilings) cases are also discussed in these chapters. Once the main objects are thoroughly introduced, the appropriate formalism for Fourier analysis in this situation is given. Because the objects of interest are infinite, this is extremely nontrivial. The payoff is that an effective mathematical version of diffraction theory can then be developed. Every major example to be found in the literature is presented and analyzed in detail.

As a mathematician who specializes in self-similar tilings and is trained in the ergodic theory of dynamical systems, I often refer to the more physics-oriented parts of the book not only for technical details but also for an understanding of the big picture. The clarity and thoroughness with which the material is presented and the concrete examples on which the theory is applied has made this an essential reference for my work. 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 is a Professor of Mathematics at Vassar College in New York. In addition to studying the mathematics of self-similar tilings and their variants, she makes art from them. Both her mathematics and her art can be seen at

Foreword Roger Penrose
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