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.
Highlights
With the concept of the neighbor type, we provide a quantitative version of the open set condition
Let us note that a fractal tiling with irrational rotations was found in [5] using a reflection and the fact that M involves an irrational rotation
Whenever we extend the self-similar construction of a connected attractor A to the outside, by forming supertiles f k−1 ( A), f k−1 f k−1 ( A), . . . , any isometry between two ‘tiles’ of such a pattern will belong to that group
Summary
A self-similar set A with pieces rotated by multiples of 60◦ , as in the middle of Figure 1, can be generated by a translationally finite graph-directed system of six sets without considering rotations. In this approach, all neighbor maps are translations, and since all matrices are powers of a basic matrix M, their product is commutative. This leads to a very convenient concept of an algebraic planar IFS which saves all matrix calculations.
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