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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 → ∞.