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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: The complex logarithm is multi-valued. To take advantage of this, the MandelbrotBC (Branch-Cut) formulas are supposed to make visible all the fractals that result from the infinity of values that may be given to the 'k' part of the logarithm. This situation exists because the formula for the complex logarithm is: (1/2)ln(x2+y2)+i(atan(y/x)+2kpi) I am not sure if any of the MandelbrotBC formulas calculate the different values of 'k' correctly, or even if a correct way actually exists. In fact, I'm not even sure of where to look to find a correct way. The correct way, if one exists, would probably fall in the field of 'complex variables' or something like that, but information on 'complex variables' is not easy to find. One thing I am sure of however is that the MandelbrotBC formulas, with the various functions applied, make some incredible fractals, so who cares if the images are mathematically correct. Today's image lies in the Z^(1-2/3)+C Mandeloid as it appears 83 levels up the logarithmic spiral, with the 'floor' function in effect. At this level, the Mandeloid somewhat resembles a distorted Mandelbrot set standing on its nose. The scene is located deep in the chaos on the western shoreline of the large period-2 bud, which is actually shaped rather like a nose. The minibrot at the center of today's image suffers from an insufficient maxiter, but its surrounding pattern is unusually well-defined for a minibrot in this range. All things considered, I rated the image at a 6-1/2, not quite as high as I would have liked, but high enough to make the calculation worth the 4-1/2 minutes required. The name "Age of Iteration" has no significant meaning. I simply pulled it out of the air. Finally, if all else fails, the completed GIF image is available on the FOTD web site at: In comparison to Tuesday, Wednesday's weather began peacefully here at Fractal Central, with singing birds and wisps of fog hanging over the river. When the fog cleared away, we were left with a day of mostly clear skies and a temperature of 72F 22C. The fractal cats purred approval, then meowed for breakfast. After such a hectic week, Wednesday, which actually was about average, seemed unusually slow. But psychological time ignored, the next FOTD will be posted in 24 hours. Until then, take care, and when it comes, recognize it for what it is. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START PARAMETER FILE=======================================
Age_of_Iteration { ; time=0:04:30.82-SF5 on P4-2000
reset=2004 type=formula formulafile=allinone.frm
formulaname=MandelbrotBC3 function=floor logmap=428
center-mag=-0.2811119334670882/-0.991563627943696/\
5.35274e+011/1/10/0 passes=1 maxiter=2000 inside=0
params=1.6666666666667/0/83/0 mathtolerance=0.05/1
float=y periodicity=10
colors=000LTdKUbIV_HWYFXVEYTDZQBZOAZL8ZJ7ZG5ZE4_B3\
`96a88`8B_8D_8F_8I_8KZ7NZ7PY7RY7UX7WW7ZV6`V6bV6eV6\
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zwkzrizmgvifrdem`ehWecRdZNdUIdPEdKFcFFcAGc5Gc6GU7H\
c8HU9HcAIUBIUCJUDJUEJUFKUGKbHKbILbJLbKMbLMbMMbNNbO\
NbPNbPPcOQdNSeNTfMVgLWhLYiKZjK`kJalIcmIdnHeoGgpGhq\
FjrFksEmtDnuDpvCqwBsxBtyAuzAswCrtEqrGpoHomJnjLqkJm\
hMjePfbSc`V`YYXV`UTcQQfNNiMLmKLkJLiILhHLfGGdFGcEBa\
DB`CBZBBXABW90U8UzUUzzUzzUzzUzdKzaKzZJzWIzTIvPHrMG\
mJFhGFeDEdAEb70_90XA0UB0UD0UEQYFSXHUVIWUJYSL_QMaPN\
cNPeLQgKRiISkHTmHUnHVoHXpHYqHZrH_sH`tHauHbvHcwHdxH\
byIczJmzKqzKvzLzzMzzMzzNzzOzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzz }
frm:MandelbrotBC3 { ; by several Fractint users
e=p1, a=imag(p2)+100
p=real(p2)+PI
q=2*PI*fn1(p/(2*PI))
r=real(p2)+PI-q
Z=C=Pixel:
Z=log(Z)
IF(imag(Z) > r)
Z=Z+flip(2*PI)
ENDIF
Z=exp(e*(Z+flip(q)))+C
|Z| < a }
END PARAMETER FILE=========================================
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