The area of the Mandelbrot set (Mset) is pretty much an inconsequential number other than the fact that it is extremely difficault to calculate with any great degree of accuracy. The ability to view the Mset in the first place is a luxury of the computer age, due to the sheer number of calculations required to draw a colour, or even black and white image.

Since the Mset is such a complicated shape, it is extremely difficult to determine the area which lies within the boundary of the set, hence the interest in the task.

There are two popular methods of determining the area of the set, - one sums the series of a sequence known as the 'Laurent series' although this converges extremely slowly. A better method is simply counting pixels - except it's not that simple. The idea is to calculate an n by n grid of pixels and count the ones for which the pixel lies inside the set. There are a number of factors which affect the calculation:

1) The number of iterations calculated before assuming a pixel lies outside the set. this is sometimes called the 'dwell limit'. Methods exist to get round this problem, by effectively having an infinite number of iterations - eg, by detecting orbits.

2) Results can depend on the offset and spacing of the grid for which pixels are to be calculated.This can be overcome to some extent by averaging out a multiple number of results at differing grid offsets.

3) Since a pixel is effectively a tiny area in its own right then depending on whereabouts in that pixel you chose to do your calculation, then that pixel may or may not really belong in the set. The problem is that those accidentally included may not necessarily balance those accidentally excluded. This problem is related to number 2. A solution may be, in the case of pixels extremely close to the boundary, to subdivide pixels into four sub-pixels and re-calculate each sub-pixel - and for each sub-pixel inside the boundary, accrediting it with an area equal to one quarter that of the original pixel. Further subdivision, etc., of each sub-pixel may also be necessary.

4) The computing power necessary to do all the calculations for a dense enough grid in order to get a reasonably accurate result is immense.Using a Silicon Graphics Indigo 2 Impact R10000, it takes many hours (albeit without some much needed optimisation) to get a result accurate to more than about four decimal places - approximately 1.5066.

My aim with respect to this challenge is to create an application which can be distributed not only across many machines, but also between many people, in order to massively distribute the calculations across the world. A 2^20 by 2^20 grid of
pixels would be a pretty big task, but not impossible, if data was collected on a line by line basis - ie the number of pixels on each line of the grid, which lie inside the set.

I can't help thinking, though, that there must be some other calculation which could be distributed in this way, and which may have more useful results once calculated...for instance, pi to n billion decimal places - very useful that!

Watch this space for more information.