Michael Baake & Uwe Grimm

— a book series on aperiodic order, published by Cambridge University Press

Volume 1: A Mathematical Invitation (August 2013)

Volume 1: 
A Mathematical Invitation  

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.

Robert V. Moody's review in the Mathematical Intelligencer
Aernout van Enter's review in Zentralblatt für Mathematik
Jean-Pierre Gazeau's review in Mathematical Reviews (MathSciNet)
Have a look at the table of contents and read Roger Penrose's foreword
Download Addenda and Corrigenda (PDF)

Volume 2: Crystallography and Almost Periodicity (November 2017)

Volume 2: 
Crystallography and Almost Periodicity  

This second volume begins to develop the theory in more depth. A collection of leading experts in the field, among them Robert V. Moody, introduce and review important aspects of this rapidly-expanding field.

The volume covers various aspects of crystallography, generalising appropriately from the classical case to the setting of aperiodically ordered structures. A strong focus is placed upon almost periodicity, a central concept of crystallography that captures the coherent repetition of local motifs or patterns, and its close links to Fourier analysis, which is one of the main tools available to characterise such structures.

The book opens with a foreword by Jeffrey C. Lagarias on the wider mathematical perspective and closes with an epilogue on the emergence of quasicrystals from the point of view of physical sciences, written by Peter Kramer, one of the founders of the field on the side of theoretical and mathematical physics.

Download Addenda and Corrigenda (PDF)

Michael Baake & Uwe Grimm