alt Hacker NewsThe 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.
Replies
SoftTalker • today at 7:11 PM
On of the first things my geometry teacher emphasized in 9th grade was that a drawing (even a very carefully measured one) didn't prove anything. Proof had to be derived from axioms and other proven facts.
ta2112 • today at 6:29 PM
This error can be made for calculating the length of any curve. If you add the deltas in only one dimension, then you end up with a bounding box length measurement that doesn't follow the contours of the curve. It's a misuse of calculus, that can be done with or without the visualization.
I'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.