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Self-affine tiles and fractals are known as examples in analysis and topology, as models of quasicrystals and biological growth, as unit intervals of generalized number systems, and as attractors of dynamical systems. The author has implemented a software which can find new examples and handle big databases of self-affine fractals. This thesis establishes the algebraic foundation of the algorithms of the IFStile package. Lifting and projection of algebraic and rational iterated function systems and many properties of the resulting attractors are discussed.
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.