Troll Physics

This actually bothered me a little bit, not because I believed it, but because I couldn’t find anything wrong with the argument, just the conclusion. My first guess as to why it was wrong was in the second step, where two smaller rectangles were removed, but I worked it out myself and they turned out to be square, so that wasn’t the problem.

I continued working on it for a while and sent it to one of my friends from graph theory, she found the flaw in it before I did, and it’s in the final step, where you “repeat the procedure to infinity”.

Repeating things to infinity requires a demonstration of the limiting behaviour of a function. In this case, the function we are investigating is a function where S is defined inductively to be:

The square of perimeter 4 is in S

If a curve C is in S, then a curve just like C but with corners indented like the above graphic (the inner corner on the right is on the unit circle) is also in S

Taking this inductive definition to be the entirety of S, it follows that S is well-ordered, totally ordered and countably infinite, and therefore a bijection between S and the natural numbers exists.

This bijection will allow us to study the limiting behavior of the function. The basis step shows that the least element has perimeter 4. Then, in the induction step, a perimeter-preserving transformation produces the next element in the set. It follows by well-founded induction that all elements in S have perimeter 4, no problem there.

Define to mean apply f composed with itself to x, n times. . Starting with an element f(x) appears more “round”, the claim from the reddit troll is that the limit of is a circle, this is the suspect part that my friend determined was bogus.

The troll’s argument is superficially similar to Archimedes’ approximation of the value of π. Archimedes used polygons inscribed in a circle to find the approximate value of pi:



Notice that at each step, the angles decrease, so the shape rounds out more, and the perimeter changes. With the troll’s argument, all of the angles involved are right angles, so as the n increases, the curve may grip the circle more closely, but it will forever remain jagged and fraught with right angles.

The limit of f as n approaches infinity is not a circle, it’s some right angled fractal beast [edit: I've been corrected, the limit is a circle, but the length of the curve is defined in terms of the derivatives, which means that it is not defined on the corners], pi does not equal 4, proceed to infinity with caution.