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Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations
- Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.
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| Author: | Christoph Bandt, Dmitry Mekhontsev |
|---|---|
| URN: | urn:nbn:de:gbv:9-opus-59362 |
| DOI: | https://doi.org/10.3390/fractalfract6010039 |
| ISSN: | 2504-3110 |
| Parent Title (English): | Fractal and Fractional |
| Publisher: | MDPI |
| Place of publication: | Basel |
| Document Type: | Article |
| Language: | English |
| Date of first Publication: | 2022/01/12 |
| Release Date: | 2022/11/01 |
| Tag: | aperiodic tile; fractal; quadratic number field; self-similar |
| Volume: | 6 |
| Issue: | 1 |
| Article Number: | 39 |
| Page Number: | 23 |
| Faculties: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik und Informatik |
| Collections: | weitere DFG-förderfähige Artikel |
| Licence (German): |  Creative Commons - Namensnennung |