This may seem pretty random, but I point you toward this post for a couple of reasons.
First, it is, to me, a superior example of explaining esoteric math in a fashion that nonscientists can understand. It is very hard — I never really know how incomplete my knowledge is until I have to explain it to readers making no claim on that knowledge. IMO, George Dallas shows how well he knows his subject.
And then, of course, there is Dallas's chosen subject, fractal geometry. One wouldn't read deeply on a subject she or he wasn't interested in, but I've been hunting for a couple of days to understand the significance of fractals in general, and especially what's known as the Mandelbrot set, a classic fractal shape. I've been fascinated by Benoit Mandelbrot, and fractals, for at least a decade, but have never known their practical value.
So here's where I try to explain what I may or may not grasp. The geometry most of us learned in high school described squares, triangles, and other regular shapes created by simple rules: Four 90-degree angles connecting four lines of equal length creates a square, for example. The rules in fractal geometry are similarly simple: Take a value. Run it through a simple formula. Take the resulting value and run it through the same formula. Repeat. Graph those values and you get complex shapes known as fractals.
What distinguishes these forms of geometry — classic and fractal — is that the former describes shapes rarely appearing in nature. But the self-similarity (pieces of the whole have the same characteristics as the whole) found in most fractals is useful for mathematically describing nature. For example, a tree that grows from trunk to bough to branch to twig in the same way — say, in two divisions at each transition — can be said to be self-similar.
That's the value of fractal geometry — it is vastly more suited to describe and model the natural world. Read more about it at George Dallas's blog. He taught me a lot of what I think I know.