Imagine that you wanted to measure the length of the coast of Australia. You could look at a map and use a ruler to measure the distance and come up with a number you feel comfortable with. After all, you trust that the map was pretty accurate and you were careful with your measurements with the ruler. But after a little thinking, you realize that the map could be a little off so you head to the Australian coast and tediously take a meter stick to measure, even if it takes you quite a long time. After an exhausting journey, you finally get an answer, which is much larger than what you first measured with the map. But then, you think for a moment. "Why stop there?" you ask yourself. You then get a toothpick and measure again, then smaller, and smaller still. The smaller the object you use to measure the coast, the longer the coast seems to become. This is called the coastline paradox and obviously is not unique to the coast of Australia.
In fact, this is true for any other object that has curves. What this is showing is that length is not an absolute property, it is dependent on how it is measured. Consider the Mandelbrot set (which can be seen as my profile picture). This fractal pattern can be zoomed in on indefinitely, and as you look closer you see that what appears to be a fuzzy circle shape actually has countless twists and turns. If you were to measure the perimeter of this fractal, it would be infinite. You would see that the more and more closely that you look at the perimeter, there is aways more there. It is also clear though, that the entire picture has a finite area. After all, the entire Mandelbrot set fractal image can fit in my profile picture.
This is interesting, because this means that it has an infinite perimeter but a finite area! A similar problem students see in calculus is something known as Gabriel's horn. It is an object that has an infinite surface area, but a finite volume. The world is filled with many fascinating paradoxes and will explore more in the next post.