I love the fractal I call the map-of-France (but which RWG can probably correct my terminology as he may have done in the past): Start with a hexagon (stage 0) centered at 0 in C, and with vertical sides 1 unit apart. Now surround that with six copies of itself and normalize this rosette by dividing the whole thing by sqrt(7) to get stage 1. Repeat ad infinitum. The thing would converge to a fractal shape with 6-fold symmetry except that the stages keep rotating a bit. So the normalizations should divide not just by sqrt(7) but by a complex number of length sqrt(7) that counteracts this tendency to rotate. (Does it work if that number is always 2 + exp(2pi*i/6), or does the rotation angle need to vary with the stages?). The result is a fractal-boundary hexagon, a rosette of 7 copies of which have the identical shape. So, the boundary B_n of stage n converges to a fractal curve B of Hausdorff dimension d(B) = log_7(9) = 1.129.... It seems as if no matter how this is done, B will not have mirror symmetry, just rotational. BUT there is a lot of choice in how 6 copies of each rosette are placed around itself, so this method can give a plethora of different curves in the limit. (Probably continuum many, given a countable number of discrete choices.) QUESTION: Among all these choices, is there some optimization problem that will result in a much smaller collection of maps-of-France, or ideally only essentially one? Perhaps the Hausdorff dimension log_7(9) *measure* of the limiting shape can be minimized? I'm fuzzy about just how to compute Hausdorff *measure* from its usual definition. --Dan Veit asks: << Are fractal sets ever the solution of an optimization problem? Sometimes the brain has a mind of its own.