Symmetry-Adapted Fourier Series for the Wallpaper Groups
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 relatio...
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| Format: | Article |
| Language: | English |
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MDPI AG
2012-07-01
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| Series: | Symmetry |
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| Online Access: | http://www.mdpi.com/2073-8994/4/3/379 |
| _version_ | 1798043529554952192 |
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| author | Bart Verberck |
| author_facet | Bart Verberck |
| author_sort | Bart Verberck |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T22:50:20Z |
| format | Article |
| id | doaj.art-d9b180f5659a4f8f953fa2a5abb8801d |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T22:50:20Z |
| publishDate | 2012-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-d9b180f5659a4f8f953fa2a5abb8801d2022-12-22T03:58:36ZengMDPI AGSymmetry2073-89942012-07-014337942610.3390/sym4030379Symmetry-Adapted Fourier Series for the Wallpaper GroupsBart VerberckTwo-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.http://www.mdpi.com/2073-8994/4/3/379wallpaper groupsFourier seriessymmetry-adapted functions |
| spellingShingle | Bart Verberck Symmetry-Adapted Fourier Series for the Wallpaper Groups Symmetry wallpaper groups Fourier series symmetry-adapted functions |
| title | Symmetry-Adapted Fourier Series for the Wallpaper Groups |
| title_full | Symmetry-Adapted Fourier Series for the Wallpaper Groups |
| title_fullStr | Symmetry-Adapted Fourier Series for the Wallpaper Groups |
| title_full_unstemmed | Symmetry-Adapted Fourier Series for the Wallpaper Groups |
| title_short | Symmetry-Adapted Fourier Series for the Wallpaper Groups |
| title_sort | symmetry adapted fourier series for the wallpaper groups |
| topic | wallpaper groups Fourier series symmetry-adapted functions |
| url | http://www.mdpi.com/2073-8994/4/3/379 |
| work_keys_str_mv | AT bartverberck symmetryadaptedfourierseriesforthewallpapergroups |