Thanks Yenchin that explained a lot. So, the diagonal zig-zag does eventually become a straight line (when the number of divisions reaches infinity), it just doesn't equal the distance of two sides of the square anymore?
So my friends and I are arguing about a theory I've come up with. Given a square, you can divide it into 4 squares, and it will the same distance from one corner to the other if you go through the big square via the sides of the smaller squares as it would going around the perimeter of the...