THE CONSTRUCTION OF SELF-SIMILAR TILINGS

Richard Kenyon

Abstract. We give a construction of a self-similar tiling of the plane with any prescribed expansion coefficient λ ∈ C (satisfying the necessary algebraic condition of being a complex Perron number).

For any integer m > 1 we show that there exists a self-similar tiling with 2π/mrotational symmetry group and expansion λ if and only if either λ or λe^{2πi/m} is a complex Perron number for which e^{2πi/m} is in Q[λ], respectively Q[λe^{2πi/m}].

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