Here’s a little something that’s puzzled me for a long time now.
In math there is a concept called an asymptote. It represents a limit that a function approaches but can never quite reach.
Limits are a very important concept in mathematics. They allow us to reason about infinity in ways we couldn’t otherwise.
Consider the common example of calculating the area under a curve. Without using calculus, the best method of doing this involves representing the area with several rectangles whose edges touch the bottom of the curve and summing the areas of these rectangles:
The greater the number of rectangles used, the more accurate the approximation.
The concept of an integral is that it provides a way of actually determining what this sum approches as the number of rectangles goes to infinity.
Pretty simple to understand, right?
So here’s what’s so odd to me. Take the following jagged shape and figure out its area:
10, right? Pretty easy. Now, see how there are 4 “steps” along the jagged side? Suppose we increase that number to 8:
The area within this shape is now 9. If we further increase the number of steps to 16, the area becomes still smaller (8.5). Without thinking too hard about it, we can see that as the number of steps goes to infinity, the area of this shape will approach that of the triangle below: (4 x 4) / 2 = 8.
But let’s go back a second. When we had 4 steps, what was the length of that jagged side?
Each “step” has two sides of length 1 each. So, 4 * 2 * 1 = 8.
And what about when we had 8 steps?
By the same reasoning, 8 * 0.5 * 2 = 8 again. Huh, weird. And for 16 steps, it would be 16 * 0.25 * 2 = 8 yet again.
Again without thinking too hard about it, it seems clear that the length of that jagged part of the shape, and thus the perimeter of the triangle to which it converges, remains constant: the jagged part stays at 8, making the perimeter 16—same as that of a square!
So basically, the two shapes below could have the same area, while one has a perimeter of 8 + 4 * sqrt(2) but the other has a perimeter of 16.
Isn’t that weird?