Isoperimetric Tower

Hi, I thought that some here might find this interesting. I was looking through the book "Geometry Revealed" by Marcel Berger - most of which, regrettably, is way beyond me - and was reading about proofs of the isoperimetric problem. He describes one that involves taking a closed, not necessarily convex, curve and gradually displacing each point normal to the curve, an amount proportional to the curvature at that point. The resulting curves will gradually approach a circle. Here, at each iteration the new curve was rescaled to the original area, otherwise they would just shrink to a point. (So, "isoperimetric tower" is really a misnomer, I know)

I tried it using Anemone and it really, really works! A param viewer shows that the ratio of the perimeter squared/area approaches 4 Pi, even with a pretty contorted starting curve.