Finally, contact with Dr. J. He just cannot resist those technical questions. Alas, he won't answer my email questions about the FractoBowl games. Since he won't say anything, we can assume his favorite team lost. Tonight, it would seem, he has been reading his email and saw something of interest. So I'm taking a break from the sequence of panoramas, like last nights panorama.
Peter Jakubowicz asked about the perimeter of the Mandelbort set lakes. I replied, in the usual way, "Been there, done that". Just kidding, got to stall somehow. But Dr. J put me to shame as only someone from that fractal space can. He just went for his evening stroll around the grounds and counted his steps. After all, isn't that the definition of a fractal perimeter. He says he can take big steps following the lower count escape contours, which should be obvious. But as he huffed and puffed around the higher iteration count contours (see Big Fatso) to see why he huffs and puffs) he reported he had to take steps about half as long for each contour. That is so he can follow the curves. He sent along the following picture, Figure 1, showing his garden pathways.
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Figure 1. Dr. J's paths around the ground of his home.
n perimeter
1 12.5663606457797931
2 8.8944950515498699
3 7.7383729769791320
4 7.4436432355331474
5 7.7000303808066779
6 8.1421088527093027
7 8.9862442676535721
8 9.9311532506364844
9 11.2267264962016062
10 12.7021596574680396
11 14.5645917174476154
12 16.6181359097436469
13 19.2297329357505035
14 22.1410545197811443
15 25.6606374585242084
The perimeter seems to be a function of n, the iteration count for the contour, namely, a one fifth power.
P = C1*2n/5
Since Dr. J had to take steps half as long to insure getting around every wrinkle in the path, we know his step lengths are
L=C2*2-n
Therefore, the dimension must be the exponent in P/L.
P/L = C*26/5
and the dimension is 6/5.
Enjoy
Jay
PS. Newton's method was used to follow the contour around the MSet. Approximate
halving the step size was required to prevent straying from the path. Following
the path to higher order is of ever more difficulty. I have reached
the limits of 80 bit floats and weekend runs on a P200. If the trend
is real, the dimension using this measure is 6/5.
PPS The trend is not real, we already know the dimension of the boundary is 2.
Dr_Js_grounds { ; A simulation of Dr. Js garden pathways.
reset=1960 type=mandel center-mag=-0.5/0/0.6666667
params=0/0 maxiter=16 float=y colors=@default.map
}
