Only countably many points "arrive" at their destination in a finite number of steps. There are uncountably many points that never "arrive," but whose limit is on the circle.
The usual picture given to non-mathematicians about sequences and limits can end up being strongly misleading in cases like this. Just because a sequence never "gets there," the limit is still the limit. This is the same kind of murky area that talks about 0.999... recurring never "getting to" 1. It doesn't have to "get to" 1 because it's never travelling.
It's also the kind of problems that arise when talking about proof by induction. Talking about dominoes falling down is, in the longer term, very misleading. We prove P(1), and we prove that P(n) => P(n+1), then they are all true. They don't become true one by one, they are simply all true - it is what it is.
I hope that helps.
I'm thinking of starting a blog to talk about things like this - it falls between the levels of the non-mathematician and true researcher.
I think I see what you're getting at, but I guess I still don't fully understand. Any thoughts on proving that the perimeter is pi without using the knowledge that the limiting shape is a circle? Does that question make sense? To me it seems strange that for any integer number N (number of times this chunking operation is applied), the perimeter is 4, but somehow in the limit the perimeter is not 4.
Edit: Nevermind, reading cousin_it's posting I think I've got a handle on it. My confusion was exactly the difference between a sequence of approximations, and the limit itself.
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.
