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Areas of Mandelbrot fractals - by Don Cross, February 2005

z' = zⁿ + c

Below is a table of areas of these fractals as a function of n, as determined by C++ software I wrote. Because I am using a fairly low-resolution pixel-counting algorithm, I am listing the areas to only 4 places after the decimal. By comparing the value of A(2) with other people's empirical results, chances are the values here are only correct to 3 places after the decimal.

Area of Mandelbrot fractal

n	A(n)
1	0.0000
2	1.5065
3	1.7939
4	1.9808
5	2.1139
6	2.2152
7	2.2947
8	2.3608
9	2.4153
10	2.4625
11	2.5028
12	2.5365
13	2.5667
14	2.5947
15	2.6182
16	2.6415
17	2.6614
18	2.6794
19	2.6968
20	2.7125
30	2.8179
40	2.8793
50	2.9183
60	2.9470
70	2.9681
80	2.9847
90	2.9983
100	3.0097
1000	3.1181
10000	3.1343
∞	π = 3.1416

The conjecture that A(∞) = π is based on the observation that the fractals become closer and closer to a unit circle as n → ∞.

References

• A Statistical Investigation of the Area of the Mandelbrot Set by Kerry Mitchell
• MathWorld's entry for the Mandelbrot Set contains the quote: " The area of the set is known to lie between 1.5031 and 1.5702; it is estimated as 1.50659...."
• Robert P. Munafo's Pixel Counting page lists an estimate of A(2) = 1.50659177 ± 0.00000008.