Mapping the aperiodic landscape, 1982–2007

Abstract

The discovery of quasicrystals galvanized mathematics research in long-range aperiodic order, accelerating the dissolution of the the ancient periodic/non-periodic dichotomy begun by Penrose, Ammann, de Bruijn, and Mackay. What does the aperiodic landscape look like now, 25 years later, and what frontiers are still to be mapped?

Keywords:

• aperiodic tilings
• aperiodic point sets
• matching rules
• diffraction
• quasicrystals

Acknowledgements

It is a pleasure to thank Michael Baake, Chaim Goodman-Strauss, Petra Gummelt, Jeffrey Lagarias, Ron Lifshitz, Robert Moody, Boris Solomyak, and the referees for helpful suggestions, and participants in the Quasicrystal Silver Jubilee for their insightful comments and questions.

Notes

Notes

1. There are fascinating parallels between mathematical crystallography on the eves of the revolutions of 1912 and 1982, but this is not the place to draw them.

2. A set of tiles is said to be aperiodic if every tiling built with copies of those tiles is nonperiodic.

3. But beware: there are subtle differences in the way different authors define their terms, and even in successive papers by a given author.

4. This number of course depends on how we define ‘distinct’ and ‘the same’. Translation-equivalent suffices for sameness for most purposes.

5. Quasi-conversely, there is a ‘natural’ tiling associated with any discrete point set (its Voronoi tiling), but that will play no role here.

6. ‘Delone’ is a transliteration from the Russian; it is sometimes written Delaunay, as the name was originally French.

7. A point is visible if it has relatively prime integer coordinates. The set of visible points in any dimension has ‘empty holes’ of arbitrarily large radius.

8. The condition takes different forms, depending on the context.

9. Using the language of measures, as developed by Laurent Schwartz (or, e.g. N.G. de Bruijn).

10. Carefully defined as limits of sequences.

11. This follows from the existence of aperiodic tiles.