Mandelbrot started thinking about the mathematical consequences of the idea of roughness when, as a young researcher, he considered the question, "How long is the coast of Britian?" Obviously, that could be measured in miles and an answer given. But, Mandelbrot thought, the smaller the scale of measurement, the more uncertainty went into the final result. The coast was not a straight line but a jagged one, zigging back and forth and adding distance to the measurement. Not only inlets and bays had to be considered, but when you dropped the scale small enough, large rocks could affect the final answer. Make it smaller, and even gravel had to be taken into account in order to be precise. In fact, the more Mandelbrot considered the problem, the closer he came to believing that there was no exact length of the coast, because the precise measurement would have to take the variations caused by subatomic particles into account, and that wasn't possible.
Dr. Mandelbrot leaves behind the idea of fractal equations and a particular equation called, in his honor, the Mandelbrot set. Mandelbrot sets create the different bizarre repeating shapes that we see in posters, album covers and T-shirts. Those drawings are actually the graphs of these strange equations. Some equations, when the answers are plotted on graph paper, make curves and are named after the shape of those curves. Thus, equations that plot out as "parabolas" are "parabolic equations," those that plot as hyperbolas are "hyperbolic equations," those that plot as ellipses are "elliptical equations," and so on. All of these are so-called smooth curves, though, and Mandelbrot brought a whole different set of circumstances to the table when he began dealing with equations that were not smooth, but were rather "rough," like things found in the real world.
Some equations do not have exact solutions and their graphs do not make simple curves. Instead, the graph is a pattern that repeats itself, usually smaller and smaller each time, but never actually finishes even though the pattern may become too small to measure even with microscopes. The equations which produce these pictures are often weird and have answers that seem to make no sense. For example, one such equation shows that there are an infinite number of numbers between 1 and 2, or 2 and 3, or any other pair of numbers. Half the numerical "distance" between 1 and 2 is 1.5. Half the "distance" between 1.5 and 2 is 1.75. Halfway between 1.75 and 2 is 1.875, and so on ad infinitum. The repetition of the equation is called "iteration," and fractals are equations that near -- but never reach -- zero as they continue to iterate. Since you can never find a pair of numbers between which you cannot slip another number, you literally have "infinity" between any two numbers you can name.
Reading a little about him at the obit and here, one learns Mandelbrot was a bit of an eccentric. That's kind of what you'd expect, though, from a guy who, if asked, "What's between 1 and 2?" might answer, "Everything."
(H/T University Diaries, which used the headline I wanted to use)