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area of the main bulb of the Mandelbrot set

The area


The main bulb of the generalized Mandelbrot set is usually the central and biggest part of the set:

Main bulb higlighted for the Mandelbrot sets with k = 2, 3, 4 and 11

In this article we seek finding its surface area.


The maths


Recall that we got the following result for boundary of the main bulb of the generalize Mandelbrot set :


We can calculate the area inside of the curve by using the Green's theorem from potential theory, that is basically the same as the Stokes theorem applied to 2D.


This basically means that we can replace an integral over a surface by an integral over the boundary of that surface. It's obviously the perfect case for what we need, because the area is basically the integral of the constant function 1 over the surface, and we indeed know the boundary curve for it. It's a classical result that by making P=-y and Q=x, the theorem transforms into

Since our curve is parametric, we can do

what leads to

with

This means that


For the standard Mandelbrot set, the area of the main cardioid is then . The first area values as a function of k are:



It's clear also that , but since we already showed that the whole set becomes the unit disk in the limit, we can conclude that the relative area of the rest of hyperbolic components vanish as the degree of the polynomial increases. This is something that agrees with the graphical experiments.



The Area of the Mandelbrot set


Much has been discussed about the area of the standard Mandelbrot set. Only one mathematically exact solution has been proposed, but it happens to be far too slow to use in any computer. The area is given as a summation series, may be a bit disappointing for many people that would expect a nice small and compact formula, probably involving a few universal constants ( pi, Feigenbaun number, the golden ratio, e, ...). There has been many experiments in other directions, though, as pixel counting approaches (sometimes assisted by the distance estimator to better bound the error).

Here we will not care that much about the exact value of the area for the standard Mandelbrot set, but the behavior of the area as k increases instead. We know the area is asymptotically approaching We also know, as discussed before, that the period one hyperbolic component influences more and more in the total area, so the graph should asymptotically behave as the formula we got for

This is the results after a low resolution ( 4096 x 4096 pixels, 50000 iterations) pixel counting method. However, the knowledge of and was used, as well as the rotational symmetry of the sets to speed up the calculation time (basically, the counting happend only in the red pixels in the image to the right). The table below shows some results from k=2 to k=31. No more than two decimals are probably correct:

N=2..7:1.5065 1.7959 1.9828 2.1167 2.2183 2.2986
N=8..13:2.3642 2.4188 2.4655 2.5052 2.5405 2.5712
N=14..19:2.5984 2.6230 2.6451 2.6650 2.6835 2.7000
N=20..25:2.7157 2.7299 2.7432 2.7553 2.7669 2.7778
N=26..31:2.7876 2.7970 2.8059 2.8143 2.8223 2.8299

So these are the areas for the main bulb. What about the area of the whole set? By using the same pixel counting strategy, these are the numbers I got for the first power of two valuef for k, with a grid of 16384 x 16384 pixels and an interation count of 100,000:

N=2,4,8:1.5065 1.9829 2.3642
N=16,32,64:2.6451 2.8369 2.9606
N=128,256,512:3.0366 3.081 3.1081
N=1024,2048,4096:3.1231 3.1313 3.1358
N=8192,16384:3.1379 3.13878

Of course the whole area approaches as k grows, just as we predicted theoretically.