Pick an integer p in {2, 3, 4, ...} = Z+ - {1}. If you add up the curves in the complex plane C given by f_n(t) = (1/p^n) exp(p^n * it), (so we don't have to worry about convergence), we get the closed plane curve f(t) = Sum_{0 <= n < oo} (1/p^n) exp(p^n * it), 0 <= t <= 2π in C. From plotting this, I know there are a few parts of the curve where f is not one-to-one. It is also clear that the curve contains a *simple* closed curve that bounds a unique minimal open set of positive area. Some questions: --------------- 1. Is the image of f locally simply connected anywhere at all?* 2. Does the image of f contain a differentiable arc anywhere at all? 3. Do the answers depend on the choice of integer p >= 2 ? —Dan ————— * A space X is locally simply connected if for each x in X, every neighborhood of x contains an neighborhood U of x such that U is simply connected: All closed loops in U can be continuously shrunk to a point in U.