The theory of chaos and theory of fractals are separate, but have very strong intersections. That is one part of chaos theory is geometrically expressed by fractal shapes.

Self-similarity is a dull subject because you are used to very familiar shapes. But that is not the case. Now many shapes which are self-similar again, the same seen from close by and far away, and which are far from being straight or plane or solid.

Bottomless wonders spring from simple rules, which are repeated without end.

One of the high points of my life was when I suddenly realized that this dream I had in my late adolescence of combining pure mathematics, very pure mathematics with very hard things which had been long a nuisance to scientists and to engineers, that this combination was possible and I put together this new geometry of nature, the fractal geometry of nature.

Pictures were completely eliminated from mathematics; in particular when I was young this happened in a very strong fashion.

The existence of these patterns [fractals] challenges us to study forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel.

Everything is roughness, except for the circles. How many circles are there in nature? Very, very few. The straight lines. Very shapes are very, very smooth. But geometry had laid them aside because they were too complicated.