For a smooth simply connected region of area A = 1, I would use something like the standard deviation s of the curvature of the boundary. This would be 0 only for a circle. For areas A <> 1, the figure should be s/sqrt(A) to scale right. For piecewise-smooth boundary curves, the figure should be the limit of smooth curves approximating the boundary in the appropriate sense. For more general boundary curves, it should be the limit of appropriately approximating piecewise-smooth curves. (This could all be expressed with no reference to smoothness, but I omit the details.) --Dan Neil wrote: ----- Is there a good measure of how irregular a region of the plane is? something that gives 0, say, for a circular disk, and 1 (or infinity) for a fractal gasket? the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is. -----