Speaking only for myself, and not being a mathematician, I love fractals for their beauty and intricacy. I don’t understand them. I understand that they are repeating patterns that get smaller and smaller and for some, anyway, go on doing so infinitely.
As a child I recognized that the beauty of nature embraced both randomness and pattern. Although I did not know the term, in many cases, that pattern was fractal. The branches of a maple tree and the veins in a maple leaf diverge at the same angle, about 41.5 degrees of a circle. Small twigs, large boughs, and veins in the leaf thus harmonize.
The fractal that made the news was the Mandelbrot curve, which maps how quickly a series goes to infinity, or its limit. It is infinitely fine: you can keep expanding the curve and going further into it, finding repeated patterns, and never reach the end. Programmers have given us the tools to generate and display them. But there are other types of fractals. There are fractal flames, waves, and something the author calls gnarls.
One site where you can enjoy them is UltraGnosis Fractal Art. You can order calendars with images of fractal leaves. They’re very appealing to the nature-lover in me.