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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Sometimes, while discussing fractals, I remark that the image on the screen is but a slice through an abstract four-dimensional object. When I say this, the question is often asked, 'where is the fourth dimension?' I then reply by stating that the fourth dimension is sideways to the inside of solid objects, and of course, the person I am talking with remarks that such a direction does not and cannot exist because there is no space for it. The real difficulty with the fourth, and all higher dimensions, is that we cannot visualize them. Early in life, we learn to decipher and organize our sensory input so that it forms in our minds the image a three-dimensional exterior world. The image works quite well because all people, (with the possible exception of the insane), have formed the same image. That world might or might not resemble our mental images of it, but what if, perhaps with the aid of virtual-reality computers, a child in his formative years were fed images of a four-dimensional world. Could that child then grow to visualize four-dimensional space and all the figures that might exist in it? Maybe he could, maybe not. We'll never know because the experiment will never be done. But I can't help but wonder whether a three-dimensional reality actually exists 'out there' or whether we learn early in life to organize our mental impressions into a world of three dimensions, and then go on to invent a mathematics that works with that three-dimensional image. I mention the fourth dimension because it is one of my favorite topics of contemplation when I settle into my ponder chair, and also because in the upcoming FOTD's I will often be discussing the fourth dimension. Today's image is part of a critical Mandel-plane of the four-dimensional Julibrot that results when 1 part of Z^2.2, 2.2 parts of Z, and C are added and iterated. The two prominent spirals are impressive, and inspired the name "The Great Spirals". But it takes more than two spirals to make a great fractal, so I could rate the image no more than an average 5. With a render time of 13 minutes, the parameter file is slow. The good news is that the completed GIF image will soon be ready for download on the two FOTD web sites at: The weather Friday here at Fractal Central started with low clouds and fog, which then gave way to hazy sun and an afternoon temperature of 81F 27C. Ignoring the high humidity, the cats approved of the conditions, and showed their approval by sitting on the porch, guarding the yard through most of the afternoon. I'm now going to try to take it easy for the rest of the day. If I'm lucky, no work will appear; if I'm unlucky, a few minor tasks might arise. Being a firm believer in Murphy's Law, I expect to be unlucky. In any case, the next FOTD will appear tomorrow around the same time at the same place. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
The_Great_Spirals { ; time=0:13:10.00--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=ident passes=1
center-mag=-0.17759792180486490/-0.451262352528888\
60/107065.9/1/172.5/-5.05322658717055617e-011
params=1/2.2/2.2/1/0/0 float=y maxiter=2400
inside=0 logmap=134 periodicity=10
colors=000LNQJMRGLRELSCKT9JT7IU2JT5IU7IV9IVBIWDIXG\
IXIIYKIYMIZOI_RH_TH`VHaXHaZHbaHbcHceHdgHdiHekHejGc\
jGbjGajG`jF_jFYjFXjFWjEViEUiETiERiDQiDPiDOiDNiCLiC\
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MWDNZ9Pa5Qd5Rc5Sc5Sc5Tb5Tb5Ub5Ua5Va5Va5Wa5W`5X`5X`\
5Y_5Y_5Z_5Z_5_Z5_Z5`Z5`Y5aY5aY5bY5bX5cX5cX5dW5dW5e\
W4dU5eW6eX7eZ7f_8f`9fb9gcAgeBgf2hiBggKgeTgcagajg_q\
fVrgZrgbshfrlpshjsedsaZtZTtVNtSHuOBuL5uI0tOEtUSs_e\
serpfnmgkjghghedhbai_ZiWWjTTjQQkNNkKKlHHmDIlEIlEIl\
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9cGAdGAeFBgEChDDiDDkCElBFmAGo9Gp8HqAIsCJtEJuGKwILx\
KMyMMzOLzQKzSJzUJzWIzYHz_HzaGzcFzeFzgEzhDziDzjCzkB\
zlBzmAzn9zo9zp8zq7zr7zs6zt5zuUzv5zw6zx6zy6zz6zz6zz\
6zz6zz6zz6zz6zz6zz6zz9zzC }
frm:MandelbrotMix4 {; Jim Muth
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j,
k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < l }
END 20.0 PAR-FORMULA FILE==================================
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