The Mandelbrot posts reminded me of something: Are fractal sets ever the solution of an optimization problem? Here's one instance I discovered a couple of years ago: Take an infinite square grid of red points and an identical grid of blue points. Now take the blue grid and rotate it by 45 degrees with respect to the red grid. The problem is to form a perfect matching of red and blue points such that the matching minimizes the sum of the squared distances between pairs. If the infinite value of the thing being minimized bothers you, do this: Use a continued fraction approximation for sqrt(2) to approximate the blue grid so that it becomes commensurate with the red grid (at periods that increase with the order of the continued fraction approximation). Now solve the optimization problem on just one (finite) period. To get the fractal, translate all the matched (red, blue) pairs so the red point is at the origin, and observe how the set of blue points evolves with order of approximation. Veit