TITLE: Why does the Mandelbrot set contain (slightly deformed) copies of itself?
QUESTION [86 upvotes]: The Mandelbrot set is the set of points of the complex plane whos orbits do not diverge. An point $c$'s orbit is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$.
The shape of this set is well known, why is it that if you zoom into parts of the filaments you will find slightly deformed copies of the original shape, for example:


I measured some points on the Mandelbrot, and the corresponding points from one of these smaller Mu-molecules. Comparing the orbit sequences it was possible to find points on each sequence which were very close - but this experiment did not really help me to understand anything new.

REPLY [2 votes]: See this paper:

McMullen, Curtis T., 
  The Mandelbrot set is universal. In The Mandelbrot set, theme and variations, 1–17, 
  London Math. Soc. Lecture Note Ser., 274, Cambridge Univ. Press, 2000.
  MR1765082 (2002f:37081)
PDF available at the author's site.

See also The significance of the Mandelbrot set.