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Symmetry 2012, 4(3), 379-426; doi:10.3390/sym4030379

Symmetry-Adapted Fourier Series for the Wallpaper Groups

1 Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, 2020 Antwerpen, Belgium 2 Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
Received: 11 June 2012 / Revised: 27 June 2012 / Accepted: 5 July 2012 / Published: 17 July 2012
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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Two-dimensional (2D) functions with wallpaper group symmetry can be written as Fourier series displaying both translational and point-group symmetry. We elaborate the symmetry-adapted Fourier series for each of the 17 wallpaper groups. The symmetry manifests itself through constraints on and relations between the Fourier coefficients. Visualising the equivalencies of Fourier coefficients by means of discrete 2D maps reveals how direct-space symmetry is transformed into coefficient-space symmetry. Explicit expressions are given for the Fourier series and Fourier coefficient maps of both real and complex functions, readily applicable to the description of the properties of 2D materials like graphene or boron-nitride.
Keywords: wallpaper groups; Fourier series; symmetry-adapted functions wallpaper groups; Fourier series; symmetry-adapted functions
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Verberck, B. Symmetry-Adapted Fourier Series for the Wallpaper Groups. Symmetry 2012, 4, 379-426.

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