If we write down area and perimeter error e_A and e_P as a function of n, in the limit of infinite n e_A is proportional to 1/n. However, e_P is independent of n, since the perimeter does not change at any step. Thus, if the perimeter was unequal to begin (clear by inspection), it does not get closer through this approximation.
The mistake is in accepting that anything that looks like a circle must be a circle. The fractal beast has area arbitrarily close to that of a circle, but it's clear once we look at e_A and e_P that it is not an approximation of its perimeter.
Pointwise, the limit is a circle. Every point on the enclosing shape gets mapped to a sequence of points. Each of these sequences has a limit, that limit is on the circle. The resulting implied mapping of the original square to the circle is a continuous bijection. By every sense that we usually talk of the convergence of lines, this enclosing shape does approach the circle in the limit.
The point is that the mapping involved is not a length-preserving mapping.
You make a good point intuitively, but I still don't quite buy your argument.
The circle is continuously differentiable along its circumference. The fractal is not---it's still got a large number of discontinuities (offhand, it looks like the number of discontinuities is equal to 2^(n-1) after n steps).
The area enclosed by this curve approaches that of a circle, but the curve is not a circle: it cannot be described by x^2+y^2=1 because its derivatives are not equal to those of same.
Every point on the limit of the fractal is within epsilon of some point on x^2 + y^2 = 1, for every epsilon > 0. So, if some point on the limit of the fractal were not on the circle, it would be some finite distance d from the circle. By choosing epsilon = d/2 you have a contradiction.
The circumference of the limit doesn't equal the limit of the circumferences.
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.
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.
If we write down area and perimeter error e_A and e_P as a function of n, in the limit of infinite n e_A is proportional to 1/n. However, e_P is independent of n, since the perimeter does not change at any step. Thus, if the perimeter was unequal to begin (clear by inspection), it does not get closer through this approximation.
The mistake is in accepting that anything that looks like a circle must be a circle. The fractal beast has area arbitrarily close to that of a circle, but it's clear once we look at e_A and e_P that it is not an approximation of its perimeter.