The area of the Mandelbrot Set has been discussed quite a few times during the years since the Mandelbrot Set’s discovery, and particularly starting in the early 1990’s. (See the area history page for details.) An accurate analytical estimate is more difficult than one would think.
Usually the area in question is taken to be the Lebesgue measure of the whole set. Given the definition of membership and the intricacy of the filaments it is a little tough to determine for certain whether the filaments contribute to the area. Therefore, many approaches such as that of Jay Hill measure just the interior.
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| | M(a+bi) dR dI
where M(x) is the membership function, defined as:
M(x) = 1, for all x in the Mandelbrot set
M(x) = 0, otherwise
This can be measured by actually locating the mu-atoms and mearuring them individually (as was done by Jay Hill), or by a statistical method such as pixel counting. Although it might appear that such a method would capture the area (if any) of the boundary, such is not necessarily the case; it depends on the subtle and complicated dynamics of the iterations in Siegel disk Julia sets, whether one’s set theory includes the axiom of determinacy, and other equally complicated things.
pixel counting and Monte Carlo are the two statistical methods of measuring the area. The best known estimate so far is 1.506591856 ± 2.54×10-8. See the pixel counting and Monte Carlo pages for a discussion and comparison, and a description of how the error term is computed.
Analytical methods use precise mathematical formulas that can give the area, albeit with much calculation. One such method is the Laurent Series method, described in 1992 by Ewing and Schober and discussed fully on its own page.
Another approach that has provided good results is the work of Jay Hill, who has enumerated (located and counted) all of the mu-atoms up to period 15 and all but one of period 16. This includes all the islands up to period 16 and whenever an island is found he also evaluates the areas of the smaller mu-atoms in that island, down to a certain minimum size.
For more information check the links on the Jay Hill page.