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z' = z^{n}+ 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* → ∞.

- 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.

© 2005 Donald D. Cross - cosinekitty {at} hotmail {dot} com