alt Hacker NewsI'm not sure I'd describe the one you linked as "perfectly similar". At least to me, there's a couple obvious problems:
- Folding the corners of a rectangle an infinite number of times doesn't make it a circle, it just means it has an infinite number of corners.
- The folded corners always make right triangles, no matter how small they are. If you put the the non-hypotenuse legs of a right triangle against a circle, no matter how infinitely small the legs are, the corner of the legs will never touch the edge of the circle: an infinitely small triangle can't have all three points be the same point (or it's not a triangle). Which means the area of the folded rectangle will always exceed the area of the circle it's mimicking, even with infinite folds.
- As the folds become smaller and smaller, the arc of the circle (relative to the size of the triangles against it) becomes straighter and straighter. Which means each successive fold scrunches up more perimeter while becoming less and less circle-like.
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I think your attempt to rebuke the proof is flawed too. The problem in your reasoning is mixing up "arbitrarily many" and "infinitely many".
There's no convergence after a finite number of steps. But at infinity, the canonical limit of this construction method is a circle. And because it is a circle, the circumference at infinity "jumps" to 2*pi. This is quite counterintuitive but perfectly legit in mathematical analysis. It's just one of many wacky properties of infinity.
Well, yes, it is false, hence there are problems.
But imagine this was a domain you weren't familiar with, you didn't know that pi != 4, you didn't know that the proof was false going into it. Could you have come up with a list of problems so quickly?
For what it's worth, I'm not sure that problems 1 and 2 are actually genuine problems with the proof. You can approximate the length of a curve with straight lines by making them successively smaller. This is the first version of calculus that students learn. Problem 3 is the crux.
These are obvious problems to someone who has studied enough math/geometry/calculus to know how one form of "adding boxes together gets a curve" and another "adding boxes together does NOT get you a curve".
There's also the intuition that the circumference of the circle must be less than the perimeter of the square, so if the perimeter of the polygon isn't decreasing as it gets closer to the circle, it doesn't approximate it better than the square itself.
I.e., the perimeter doesn't approach the circumference in value because it doesn't change.
It's an interesting thing to think through though, and maybe a good point about how arguments can seem intuitive at first but be wrong. On the other hand, I'm not sure that's any more true of visual proofs than other proofs.