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 square. Now, if you divide this square into smaller squares (16, for the sake of argument) it will take 8 side lengths to get from one corner of the big square to an opposite corner if you go through the outside of the big square or the inside of the big square. So no matter how many time you divide a square, the distance along the "zig-zagged" line from one corner to the other will be the same as half of the perimeter of the square. Now I said that when a square is divided into an infinite amount of smaller squares, the "zig-zagged" line through the square would become a perfectly straight line, but would still be the same length as two of the larger square's sides. Obviously this is a paradox because in a right triangle the legs have to add up to a greater value than the hypothesis. Can anyone verify if I'm imagining this right? At an infinite number of divisions, will a line form from a corner to the opposing corner, and will it be the same distance as half of the perimeter?