Rhapsody on the proof of pi = 4

Say you're me and you're in math class, and you're supposed to be graphing functions as if there were some deep relationship between y and x that your teacher just won't stop gossiping about. But like most gossip, you really don't care about y's unhealthy dependency on x. Really y, get a life. Luckily, your friend passes you a note. You wait until the teacher is facing the board, and sneakily open it up. And it's this proof of pi equals 4 that you've seen like a billion times on the internet. But because you're bored, and at least it's not graphing, you'd take another look. It's like this. Start with a circle of diameter one and circumference pi. Then draw a square around it. The length of a side of the square is 1, so the perimeter of the square is of course, 4. Now, you start approaching a circle by zigzagging this parameter. This shape still has perimeter 4, and we do it again, and it's still 4, and again. And we approach a circle while the perimeter never changes. So therefore, pi equals 4. Obviously, there's something wrong here. The circumference of a circle is pi, but pi is more like 3.14 numbers, and less like 4. So somehow what this process approaches, is something that looks like a circle, but is not a circle. I mean, what does circle mean anyway? There's this loopy line thing, And then there's this solid disk shape. And while they're related and all, they're not actually the same mathematical object. This troll proof is so much fun because, as you repeat this zigzag move, all the way to zig-finity, it approaches the shape of a circle in the disk sense, and it approaches the area of a circle, but it does not approach a circle. It's all crinkled up. And you imagine that if you stuck a straw in this thing and inflated it-- that is, added as much area as you can while keeping the perimeter as 4-- all the infinite wrinkles would smooth out, and you'd get a circle with perimeter 4 and diameter 4 over pi. In fact, you could inflate the square to get the same thing. Probably has something to do with how circles are the shape with the most area possible given their perimeter, and why soap bubbles like to be spheres, and raindrops are actually pretty sphere-ish too. But anyway, you decide the best way to respond to your friend is to try applying the same fake proof process to something else. Maybe you can choose another irrational number, like square root 2. In fact, square root 2 would work great, because it's also a common geometry number. The ratio of the diagonal of a square to its perimeter. I mean, like, if this square has side length 1, then you've got this right triangle. And a squared plus b squared equals c squared. In fact, that's how you decide to start your proof for your friend. Take a right isosceles triangle. Each leg has length 1. So the hypotenuse is square root 2. Put this all in the square of perimeter 4. Now we can approach a triangle in the same way. And each time, the perimeter of the whole shape is still 4. And thus, by the time we get to a triangle, the length of the hypotenuse must be 2. Thus, the square root of 2 is 2. So you pass that over to you friend. Will he notice that you're approaching the area of a triangle without approaching a triangle? That a triangle has three sides, but the shape you end up with is a polygon with infinite sides? An infini-gon? But not a boring one, like how a regular polygon approaches a circle as sides go to infinity, because infini-gons are more fun when there's actual angles between sides. Zigzagging along to make what you've decided to call a zig-fini-gon. You begin to wonder what other zig-fini-gons you can make. Maybe if you started with a star, and zigged all the points in. And did that again and again to infinity. You get something that looks like a pentagon, but it's actually a zig-finite star with the same perimeter as the original star. Or maybe you could have a rule where at each step you zig down only part way. And then your zig-fini-gons will have more texture to it. Maybe you could throw some zags in there too. There's something fractal-ly about it, except the perimeter never changes. You could do that to a triangle, or maybe make a square that turns into a zig-fini-square. Uh oh, teacher's walking around. Better draw some axes, and pretend to be doing math. So you turn the idea sideways, and start at zero. Go to y equals 1 at x equals 1, then back to 0 at x equals 2. The next iteration is like folding this point down to 0. So the function zigzags from 0 to 1/2, to 0 to 1/2, to 0. The next step brings the 1/2's to 0. And now the highest points are at y equals 1/4. Each step brings the highest points down to 0, and the new highest peaks are only half as high. And each step keeps the total length exactly the same. So what happens when you do this to zig-finity? In one way, it approaches this line, the x-axis, y equals 0. If there were any peaks, they'd get folded down to 0. Therefore, there can't be any. Yet, at each step we have twice as many peaks. So how can there be an infinite number of peaks and no peaks? You might reconcile this by saying that all infinite peaks are equally at zero, since all peaks get brought to zero eventually. But if everything's at y equals 0, you have just a line segment of length 2. That doesn't make sense. The length of the zigzag stays the same at each step, and at the beginning it's like two hypotenuse of two right triangles, so 2 square root 2. And 2 does not equal 2 square root 2. Another problem is that only peaks ever get brought to 0, but not all numbers can be peaks. Any fraction of a power of 2 will be a peak at some iteration, but a number like 1/3, or an irrational number, will never be a peak or a zig or a zag. So they must all be between the zigs and the zags. But there can't be any length between zigs and zags, or else that would create a peak that would have been folded down. Somehow it has to be infinitely zigzagged in a way where there's no actual line segments of any length, but only zigs and zags. Yet, there must be an infinite number of numbers between each zig and zag to fit all the irrationals in, and somehow all the pieces of 0 length add up to be something that does have length. You could imagine grabbing the ends and stretching it out, accordion style, into a line of length 2 square root 2. And then I suppose you could crumple it all back down until the whole length 2 square root 2 line is folded up into a single point.