Suffice it to say that many natural objects approximate fractals to a high degree. Natural objects also tend to be stochastically self similar and may even appear to be (almost) scale invariant. This fractal or quasi-fractal property of nature gives a nice explanation for such things as why small animals use more energy per weight than larger ones, or how brain sizes scale with body size.
Even our experiences are self-similar. I can read a race account from a pro race and it feels remarkably similar to my race experiences, although I perform at a much lower level of "fitness." Once I read an Ironman race report from a guy I did not know and it was so similar to my race experiences that I looked up who he was and what his finishing time was. Surprisingly enough he turned out to be quite young (20-24) and his finishing time was just below 9 hours.
Compare that to my age group (50-54) and my best finishing time at just above 11. Yet almost everything he described, including the remarks he made about competitors around him could have been written by me. Everything that is except the "absolute value" of the numbers. Is that self-similar or what?
Today I ran a 10K in the hills.