Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

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.

Elementary fractal geometry. 4. Automata-generated topological spaces

“…. , (8,9), and the loop at state right gets the label (9, 0). Similarly we can describe the number system with respect to any positive integer base.…”

“…Usually, the f k are assumed to be linear maps or even similitudes. As in [6,8,9] we assume that the f k are similitudes with the same similarity ratio r < 1. Compositions of the f k are expressed by words u ∈ D n as f…”

“…The next example shows that we probably can extend this conjecture to fractals which are doubly connected, that is, there are two disjoint paths between any pair of points. In [9] we studied a connected self-similar set X with Cantor sets f i (X) ∩ f j (X) for which the IFS was not unique. For disconnected sets like Figure 3 or trees like Figure 4 it is clear that there are many different representations by similitudes.…”