When people ask me what's my field? I say, on one hand, a fractalist. Perhaps the only one, the only full-time one.

If you have a hammer, use it everywhere you can, but I do not claim that everything is fractal.

The Mandelbrot set is the most complex mathematical object known to mankind.

Regular geometry, the geometry of Euclid, is concerned with shapes which are smooth, except perhaps for corners and lines, special lines which are singularities, but some shapes in nature are so complicated that they are equally complicated at the big scale and come closer and closer and they don't become any less complicated.

Nobody will deny that there is at least some roughness everywhere

If you look at coastlines, if you look at that them from far away, from an airplane, well, you don't see details, you see a certain complication. When you come closer, the complication becomes more local, but again continues. And come closer and closer and closer, the coastline becomes longer and longer and longer because it has more detail entering in.

In mathematics and science definition are simple, but bare-bones. Until you get to a problem which you understand it takes hundreds and hundreds of pages and years and years of learning.