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 connected...

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Bibliographic Details
Published in:Fractal and fractional 2022-01, Vol.6 (1), p.39
Main Authors: Bandt, Christoph, Mekhontsev, Dmitry
Format: Article
Language:English
Subjects:
Algebra
Angles (geometry)
aperiodic tile
Carpets
Crystallography
fractal
Fractal geometry
Fractals
Geometry
Number theory
Porous materials
quadratic number field
self-similar
Self-similarity
Symmetry
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Summary: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.
ISSN:2504-3110
2504-3110
DOI:10.3390/fractalfract6010039