SMTPUSER@GACVAX1.BITNET (01/13/90)
About two months ago, I send a query to the list regarding the area of the
Mandelbrot Set. Well, the answer seems to be 1.508 to four significant digits.
This is according to Dave Rabenhorst in The Journal of Chaos and Graphics,
vol. 2 August 1987. He uses a brute-force approach to finding this result.
Basically, he took a typical Mandelbrot calculating/displaying program and
modified it to count the number of points that fall in the set and dividing
by the number of points in the square (-2,2)-(2,-2) that surrounds the set.
Observing the table of values that he constructed it seems to me that, for a
particular resolution (# of points on a side) of the square around the set,
as the number of iterations increases, the number of points that fall into
the set seems to come to some limit. Since the area of the M-Set is probably
a limit of some sort, a mathematical relationship might be derived to calc.
the area.
For anyone interested in estimating the area themselves, but not wanting to do
a lot of thinking, the formula he used was:
16*N/(R*R)=Area
where N=Number of points found to be in Set for some iteration limit
and some R
R=number of points on a side of the square (-2,2)-(2,-2) that
surrounds the Set
Note: I just did a little calculation on my calculator. Using Mr. Rabenhorst's
table I found this:
Resolution Apparent limit for N (as iterations -> infinity)
33 106
65 396
129 1556
257 ~6200
513 ~24750
And the formula:
Limit of next resolution in table = 4*(Limit of one res. in table)-28
seems to predict this very well:
R Limit by formula
33 106 (assumed as a starting point)
65 396
129 1556
257 6196
513 24756
A coincidence? Probably. The formula seems to not work for the next iteration
(1025).
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Christopher Kane
CHRISTOPHER@GACVAX1.BITNET
CHRISTOPHER@GACVX2.GAC.EDU (soon) (Pithy message deleted)
Gustavus Adolphus College
St. Peter, Minnesota USA
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