yrs ago when I asked about this elsewhere, there was an assertion that "fractals are not technically defined". think there is some validity to that. since then Ive been thinking that maybe fractals are best defined as a computational sequence or "computation tableau" eg via a turing machine. this would be a very broad definition but seems to fit & dont see obvious alternative. (worthy of another question here maybe) or at least there seem to be multiple plausible definitions of fractals.
also in line with this def of fractals in terms of computational tableaus, one major contribution to study of fractals is via cellular automata patterns, heavily studied by Wolfram in New Kind of Science.
 New Kind of Science/wikipedia
 New Kind of Science/Wolfram online version