All the retards claiming the curves don't converge to a circle need to immediately stop posting on this board and not come back until they've read a book on calculus. It is easy to see (though difficult to actually prove because this process is difficult to formalize) that as a sequence of parametrized curves R->R^2, the curves described here converge (even uniformly) to one whose image is a circle. The issue here is that arc-length is defined based on derivatives, but uniform convergence of a sequence of functions is not enough to allow you to interchange a limit with a derivative operator. A sufficient condition to make that swap is uniform convergence of the sequence of derivatives plus convergence at at least one point of the sequence of functions. Unsurprisingly, that isn't even working because the sequence of derivative here does not even converge pointwise a.e..

By contrast, the classical Greek method using regular polygons works because the sequence of derivatives converges uniformly. Well, the corners are a little bit of a wrinkle, but it's converging in L^\infty which is enough for our purposes.