here's an easy way to demonstrate why this proof is unsound. let's try see if we can approximate the length of a line segment with sine waves of increasingly smaller amplitudes.

consider the following sine function, y=4sinx, and let's restrict the domain from x=0 to x=6.28.

it's obvious just from looking at the graph that the length of the sine wave is greater than the length of the domain (6.28 units). and, math-doing-robots confirm that the length of that line is

~17.628. it's also obvious that if we want to approximate the length of the domain, then we must decrease the amplitude.

next we're going to add more sine functions to the graph. the pi=4 proof demands that the perimeter of the square remain constant by changing its shape in a specific way. likewise, we're going to keep the length of the sine wave constant while we decrease its amplitude. the only way to do that is to increase its period proportionally. in other words, if we decrease the amplitude by half, then we must increase the period by half. if i'm not making sense, just check out the following graph. this is y=4sinx, y=2sin(2x), y=sin(4x), y=.5sin(8x), all from x=0 to x=6.28

if you plug all those formulae into the math-wizard-robot, it will confirm that they all have the same length, ~17.628. but now we have a problem. as you can see, we can keep iterating and the sine wave will get smaller and smaller and smaller and smaller until it

*appears* to be approximating the length the line, but since ~17.628 != 6.28, we know that it never does.

in fact, this notion of keeping the length of the sine wave constant by only letting amplitude vary inversely proportional to period is exactly what your proof does. just look at the corners. each time they "fold" the perimeter in the corners, they're doing it in a specific way that keeps the length the same, halves the amplitude, and doubles the period. graphing the absolute values of the same sine functions from before illustrates this. each iteration, starting with purple, has half the amplitude and double the period of the previous iteration, but their lengths are all the same. it might appear that they would approximate the length of a line as the amplitude approaches zero, but it never does, and they never do.

/total thread derailment