TY - JOUR

T1 - Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

AU - Bandt, Christoph

AU - Mekhontsev, Dmitry

N1 - Publisher Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/1

Y1 - 2022/1

N2 - 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.

AB - 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.

KW - Aperiodic tile

KW - Fractal

KW - Quadratic number field

KW - Self-similar

UR - http://www.scopus.com/inward/record.url?scp=85123785663&partnerID=8YFLogxK

U2 - 10.3390/fractalfract6010039

DO - 10.3390/fractalfract6010039

M3 - Article

AN - SCOPUS:85123785663

VL - 6

JO - Fractal and Fractional

JF - Fractal and Fractional

SN - 2504-3110

IS - 1

M1 - 39

ER -