Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Christoph Bandt, Dmitry Mekhontsev

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Article number39
JournalFractal and Fractional
Volume6
Issue number1
DOIs
Publication statusPublished - Jan 2022
Externally publishedYes

Keywords

  • Aperiodic tile
  • Fractal
  • Quadratic number field
  • Self-similar

OECD FOS+WOS

  • 1.01.PO MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

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