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## Areas of Mandelbrot fractals - by Don Cross, February 2005

I am currently investigating the areas of the Mandelbrot fractals formed by iterating the formula
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.

 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 → ∞.