Quasicrystals

Abstract

Mathematicians have been interested in non-periodic tilings of space for decades; however, it was the unexpected discovery of non-periodically ordered structures in intermetallic alloys which brought this subject into the limelight. These fascinating materials, now called quasicrystals, are characterised by the coexistence of long-range atomic order and 'forbidden' symmetries which are incompatible with periodic arrangements in three-dimensional space. In the first part of this review, we summarise the main properties of quasicrystals, and describe how their structures relate to non-periodic tilings of space. The celebrated Penrose and Ammann-Beenker tilings are introduced as illustrative examples. The second part provides a closer look at the underlying mathematics. Starting from Bohr's theory of quasiperiodic functions, a general framework for constructing non-periodic tilings of space is described, and an alternative description as quasiperiodic coverings by overlapping clusters is discussed.

This article is made available for reference. It was written following the March 2006 workshop "The World a Jigsaw: Tessellations in the Sciences" at the Lorentz Center, and due to be published in a book entitled "Tessellations in the Sciences: Virtues, Techniques and Applications of Geometric Tilings" (eds R van de Weijgaert, G Vegter, J Ritzerveld and V Icke), but annoyingly this never happened