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.
Keywords
Cite
@article{arxiv.1906.10392,
title = {Quasicrystals},
author = {Uwe Grimm and Peter Kramer},
journal= {arXiv preprint arXiv:1906.10392},
year = {2019}
}
Comments
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