| Pixel Counting |
Robert P. Munafo, 2003 Sep 25.
Currently (as of fall 2003) the best pixel-counting estimate for the area is 1.506 591 77 +- 0.000 000 08.
The best estimate for the center of gravity is -0.286 768 44 +- 0.000 000 025.
These estimates were produced by this program, a brute-force pixel-counting program that properly handles the sources of error described below. It counted 20 grids each with approximately 262144 by 131072 pixels, with a dwell limit of 134,217,728.
It is not known whether either the area or the center-of-gravity has an exact value computable by a simple formula, but if they do the best candidate formulas appear to be:
sqrt(6 . pi - 1) - e ?= 1.5065916514855032852705345... ?? hypothesis
for the area (found by Cyril Soler), and (far less likely):
-((ln(3) - 1/3)F) ?= -0.2867682633829350268529586... ?? hypothesis
for the center of gravity, where F is the Feigenbaum Constant. This second formula is now considered wrong based on the margin of error in the pixel-counting estimate, but there is also the chance that it is just an error in the pixel-counting.
Pixel Counting is susceptible to the following sources of error:
The features which most often cause aliasing errors are R2F(1/2B) (colloquially called the Spike) and R2.C(1/2) (known better as Seahorse Valley). Because the Spike is horizontal and Seahorse Valley is vertical, the grid can be easily aligned in such a way as to cause aliasing errors.
To determine the actual amount of error from aliasing and statistical sampling, it is best to compute a mean and a standard deviation from many seperate runs. In order for the standard deviation to be an accurate measure of error, the runs should all differ from each other by having slightly different grid spacing, being shifted slightly along real and imaginary axes, etc.
Here are the results of a set of such averaged runs. For each grid size, 20 runs were computed and averaged together. The standard deviation measures how much (on average) each run differs from the average. The average itself varys even less from the true area.
See also Laurent series and Monte Carlo, the other methods of estimating the area of the Mandelbrot Set.
See the Area History page.
Formula for Mandelbrot area: Cyril Soler.
Area estimate (Pixel counting method): Jay Hill
Old USENET articles: G. A. Edgar (edgar at math ohio-state edu)
This work is licensed under a
Creative Commons Attribution 2.5 License
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