I think I've wondered about this before but finally googled around to
see if someone else had looked into it. Here's the question: what is the
area of the Madelbrot Set? (Or, is this even a legitimate question?!)

Turns out, it is, and the answer is approximately 1.506484. I leave it up
to those of you intrigued by such a question to find the paper that
comes up with this very accurate estimate. From the article we read :

The Mandelbrot set (M) has been called the most complex object in
mathematics, and continues to be the subject of active research. One
open question is, what is the area of M? It is well known that the set
is bounded by a circle of radius 2, centered at the origin of the
complex plane. Thus, the area is certainly less than 4pi, or
approximately 12.6. Indeed, the area is much less than that. The
left-most extent of the set ends with the spike at x = -2, and the right
side extends out to approximately x = 0.47. The top and bottom are at
approximately y = +/- 1.12, respectively. This bounding rectangle has an
area of about 5.5, and even this is a gross overestimate, as shown.
Here, M is rendered in a binary fashion: points inside the set are
colored black and points outside the set are white.

The general approach used by the fellow who came up with this
approximation was to compare the count of the number of black pixels and
the area in pixels of the surrounding rectangle.



Ed Davis wrote a program in tinyC that plotted this set. (The fact that
you can even *think* about doing this is almost as awesome as the set
itself. But I digress ...)



Couldn't we modify Ed's program to count the number of M's, m's and "'s?

Anybody want to try?

http://primepuzzle.com/not.just.tiny.c/mandel.tc