That doesn't follow from what I wrote. Every shape in the sequence of shapes has a circumference/length of 4, and is not a circle. The limiting shape is a cir

That doesn't follow from what I wrote. Every shape in the sequence of shapes has a circumference/length of 4, and is not a circle. The limiting shape is a circle and does not have a circumference/length of 4, but of pi. We can have a bijection from the original square to the circle. Why is this a surprise? It's not length preserving. We can have a bijection from the interval [0,1] to the interval [0,2]. That's not length preserving either. If I've misunderstood you then perhaps you could explain your thinking in more detail. I don't understand where you think there are two versions of a circle.