Cantor's Dust is a famous fractal, a basic pattern that repeats itself over and over. It's a pretty pattern, but it didn't seem very useful at the time it was invented. Years later, it was invoked again at the dawn of chaos theory to explain an odd phenomenon in broadcasting.
The Cantor Set is the result of a complicated-looking drawing that describes a very basic function. It was invented by Henry John Stephen Smith, and put forward by Gregori Cantor in the mid-1800s. (In accordance with ironic science tradition, no idea cannot bear the name of the scientist who actually discovered it, which is why it's not called the Smith Set.)
The way to construct the Cantor Set is simple. Draw a line segment. (For practicality's sake, make it quite long.) Then go slightly down the page, and redraw the thing, but take out the middle third. You now have two equal-length line segments slightly apart from each other. Go slightly down the page, redraw both of them, removing their middle thirds. Just keep going.
In theory, this could go on and on forever. As you hack up the line segments into smaller and smaller parts, you get what's called Cantor Dust. There are infinitely many little points (after many iterations), and all of them have a length of zero. At least that's the idea.
This drawing is a fractal, a pattern that will look the same no matter how many times you zoom in or out. In an ideal model, there will always be a big, unbroken line at the top, and there will always be an infinite number of small lines at the bottom.
This was pretty but of limited use until Benoit Mandelbrot, working in the 20th century, noticed something odd about how information was transmitted over the phone. Mandelbrot was working for a little computer company called IBM, and they had the idea that transmission of information from one computer to another over phone lines might be valuable. This is why the "something odd" that Mandelbrot noticed was a problem. There would be little moments when the line developed static-y errors, and then stretches of time when it would be clear again. They wondered if there was a pattern to it, and if it could be corrected.
Mandelbrot came up with a pattern, just not the one that anyone was hoping for. There were seconds at a time when the electric signals would cloud, becoming unintelligible, and times when they would be clear. Looking within the clusters of errors, Mandelbrot saw more little clusters of errors, and more times when the signal was clear. Looking in still further, he saw more little clusters, and more times when the line was clear. No matter how far anyone "zoomed in," there was never a perfectly clear signal or an entirely interrupted signal. The shorter spaces of time could not be distinguished from the longer spaces of time. As odd and contrived as fractals seemed to be, humans had just created a fractal, by accident. Too bad it was one they didn't want.