Aperiodic order. Volume 1. A mathematical invitation. (English) Zbl 1295.37001

Baake, Michael; Grimm, Uwe

Encyclopedia of Mathematics and its Applications 149. Cambridge: Cambridge University Press (ISBN 978-0-521-86991-1/hbk). xvi, 531 p. (2013).

Aperiodic order is a subject of interest to Biology (for example E. Schrödinger already in the 40’s in [What is life? The physical aspect of the living cell. With ‘Mind and matter’ and autobiographical sketches. Reprint of the 1992 edition. Cambridge: Cambridge University Press (2012; Zbl 1254.01052)] refers to aperiodic crystals as a model for genetic information), Chemistry (in which Dan Schechtman received the 2011 Nobel Prize), Physics and also Mathematics. In some sense the subject became more unified after the discovery of the existence of a quasicrystalline phase, but it then turned out that there were many and quite varied mathematical predecessors for the description of aperiodic systems, originating from a variety of mathematical subjects.
Examples thereof are Yves Meyer’s work, originating from Harmonic Analysis, the aperiodic tiles of Robinson, Wang and Penrose, the nonperiodic Thue-Morse, Fibonacci and Rudin-Shapiro, period-doubling and paperfolding sequences. Some of those tilings and/or sequences are obtainable from the irrational projections of higher-dimensional periodic lattices, some by inflation rules, or tiling-rules (nearest-neighbor exclusions). Also the notion of order can be interpreted in different ways; a majority consensus arose in which “order” was interpreted as the occurrence of some singular component in some spectrum, experimentally measurable by diffraction experiments. (Others stilll advocate different notions, such as zero-entropy-density behaviour). The structures under consideration, usually infinite-particle configurations in infinite space, were generalised to measures on Euclidean space.
Baake and Grimm have done the community a great service by collecting and unifying the various descriptions and results about the mathematics of aperiodic order. They wrote a scholarly tome, with (as is necessary in the subject) a large number of attractive and helpful illustrations. As indicated above, the unity of the subject resides more in the physics/chemistry than in the mathematics of the topics so they had a hard task to perform of which they acquitted themselves excellently. Most of the book treats what physically corresponds to ground-state questions, in the last chapter the situation in which thermal randomness may play a role is discussed; this is as yet still an only partially developed area.

Reviewer: A. C. D. van Enter (Groningen)

MSC:

37-02	Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
00-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general
82-02	Research exposition (monographs, survey articles) pertaining to statistical mechanics
28D05	Measure-preserving transformations
42-02	Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces

Keywords:

aperiodic order; quasicrystals; tilings; substitutions; inflation; cut-and-project; diffraction; ergodic spectra

Citations:

Zbl 1254.01052