Here is another way to see what is happening, that may be more clear from an intuitive sense: From the puzzle, name the square object S and the circle inside i

Here is another way to see what is happening, that may be more clear from an intuitive sense: From the puzzle, name the square object S and the circle inside it C_1. Imagine another circle C_2 that circumscribes the square S from the puzzle. i.e. the corners of the square lie on the circle C_2. Then for each step, when we invert the outer-most corners of the square, we constrict C_2 such that the circle lies on the new outer-most points of S. What happens is that as you repeat this process more and more the outer circle C_2 gets smaller and smaller, approaching the size of the original circle inside the square, C_1. Also you can infer that the area of S is equal to the area of C_1 and C_2 since (area C_2) -> (area C_1) and (area C_1) <= (area S) <= (area C_2). Which makes sense intuitively, too, since they all enclose the same space. This tells you nothing about the relationships between the circumference of the objects, though.