This paper presents two methods for generating Koch curves, analogous to the commonly used iterative methods for producing images of Julia sets. The attracting method is based on a characterization of Koch curves as the smallest nonempty sets closed with respect to a union of similarities on the plane. This characterization was first studied by Hutchinson. The repelling method is in principle dual to the attracting one, but involves a nontrivial problem of selecting the appropriate transformation to be applied at each iteration step. Both methods are illustrated with a number of computer-generated images. The mathematical presentation emphasizes the relationship between Koch construction and formal languages theory.
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