Abstract
The symmetries of periodic structures are severely constrained by the crystallographic restriction. In particular, in two and three spatial dimensions, only rotational axes of order 1, 2, 3, 4 or 6 are possible. Aperiodic tilings can provide perfectly ordered structures with arbitrary symmetry properties. Random tilings can retain part of the aperiodic order as well the rotational symmetry. They offer a more flexible approach to obtain homogeneous structures with high rotational symmetry, and might be of particular interest for applications. Some key examples and their diffraction are discussed.
Highlights
Motivated by the discovery of quasicrystals in intermetallic alloys by Dan Shechtman in 1982 [1], the investigation of aperiodic tilings and points sets has become an increasingly active area of mathematical research
One of the key fundamental questions from crystallography is how to properly define the concept of order, which is challenging, as can be seen from the ongoing discussion concerning the definition of a crystal; see [2] for background
Singular continuous components did not play a major role, but they have become more prominent in the discussion of aperiodic structures, because they are prevalent in substitution based structures displaying some self-similarity properties
Summary
Motivated by the discovery of quasicrystals in intermetallic alloys by Dan Shechtman in 1982 [1], the investigation of aperiodic tilings and points sets has become an increasingly active area of mathematical research. Pure point diffraction is seen as the hallmark of an ordered structure, while continuous components, and in particular absolutely continuous components, are considered a sign of disorder. Aperiodic tilings and point sets serve as models for the atomic structure of quasicrystals, but are becoming increasingly important as metamaterials. The possibility to realise higher symmetries than in the severely In this brief note, we shall discuss some paradigmatic planar structures, focussing on their symmetry and diffraction. For mathematical and crystallographic background on the diffraction of aperiodic structures we refer to [7,8,9] and references therein
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