This article is from the Fractal FAQ, by Ermel Stepp email@example.com with numerous contributions by others.
Most of what you could want to know is in Branner's article in _Chaos
and Fractals: The Mathematics Behind the Computer Graphics_.
Note that the Mandelbrot set in general is _not_ strictly self-similar; the
tiny copies of the Mandelbrot set are all slightly different, mainly because
of the thin threads connecting them to the main body of the Mandelbrot set.
However, the Mandelbrot set is quasi-self-similar. The Mandelbrot set is
self-similar under magnification in neighborhoods of Misiurewicz points,
however (e.g. -.1011+.9563i). The Mandelbrot set is conjectured to be
self- similar around generalized Feigenbaum points (e.g. -1.401155 or
-.1528+1.0397i), in the sense of converging to a limit set. References:
1. T. Lei, Similarity between the Mandelbrot set and Julia Sets,
_Communications in Mathematical Physics_ 134 (1990), pp. 587-617.
2. J. Milnor, Self-Similarity and Hairiness in the Mandelbrot Set, in
_Computers in Geometry and Topology_, M. Tangora (editor), Dekker,
New York, pp. 211-257.
The "external angles" of the Mandelbrot set (see Douady and Hubbard or
brief sketch in "Beauty of Fractals") induce a Fibonacci partition onto it.
The boundary of the Mandelbrot set and the Julia set of a generic c in M
have Hausdorff dimension 2 and have topological dimension 1. The proof
is based on the study of the bifurcation of parabolic periodic points. (Since
the boundary has empty interior, the topological dimension is less than 2,
and thus is 1.) Reference:
1. M. Shishikura, The Hausdorff Dimension of the Boundary of the
Mandelbrot Set and Julia Sets, The paper is available from anonymous ftp: