The problem with visual proofs is that there are perfectly similar looking proofs that are false: https://math.stackexchange.com/questions/12906/the-staircase...
They’re great and cool for things you already know to be true, but they can be tricky.
They’re great and cool for things you already know to be true, but they can be tricky.
- 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.
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.
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.
No, but if I didn't know anything about the domain, literally any proof (correct or incorrect) would seem fine. But then it's not really "proving" anything. Knowing enough for the proof to make sense but still unconditionally accepting assertions like "if you fold the corners an infinite number of times, it makes a circle" strikes me as odd.
> 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.
But that's not what's happening here: the lines are straight, but you'd approximate the length of the curve with the hypotenuses, not the legs of the folds. Surely as you repeat this process you wouldn't think "wow, the circumference of this circle is actually equal to the perimeter of the original square." You'd have to disbelieve your own eyes and intuition and knowledge of circles to accept that this is true and hopefully you'd think "maybe I'm doing this wrong."
That's not to say 1 and 2 alone prove the visual proof incorrect, but they demonstrate that it is doing something wrong. Proofs that are correct don't have inconsistencies.
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.
I kind of ran into this when I was in high school and was introduced to limits.
For me the quandary was a "stair step" shape dividing a square with length of side "s" ("stairs" connecting two opposite diagonal corners). You could increase the number of steps—they get smaller—but the total rise + run of the stairs remains the same (2s). At infinity I reasoned you had a straight, diagonal line that should have been s√2 but was also still 2s in length.
At the very least you can say that the volume enclosed approached that of a right triangle (at infinity) but the perimeter stays stubbornly the same and not that of a right triangle at all.