Shapes of Polynomial Julia Sets

Would you like to find a polynomial whose Julia set looks like a car? a triangle? a cat? your favorite aunt? I will tell you how to construct such a polynomial. In joint work with William Thurston, I proved that any Jordan curve in the plane can be approximated arbitrarily well in the Hausdorff topology by the Julia sets of polynomials. Finite unions of disjoint Jordan curves can be approximated by the basins of attraction of rational maps. I will discuss these results and show some pictures of neat Julia sets.