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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: My first impression of today's fractal scene was one of a mass of dense alien foliage, perhaps starting to turn yellow with the approach of alien Autumn. Then, when I saw the Mandelbrot midget in the midst of the brush, I named the image "Midget in the Woods". The most unusual feature of the 8-rated image may be the totally different nature of the areas north and south of the midget. North of the midget, the surroundings appear solid, like a more or less normal fractal scene. But to the south, the features resemble an archipelago of islands interlaced with open areas of water. The dividing line cuts straight through the center of the midget. This dividing line is explained by examining the parent fractal, which is a circular open hole filled with fractal debris. The chains of debris extend beyond the open area and continue well into the solid area surrounding the hole. The size of the hole is dependent on the bailout, which is specified by the imag(p3) parameter. The greater the bailout value, the larger the hole. Knowing that the size of the hole varies with the value of the bailout, I found it simple to take a reasonably interesting midget on the outside of the open area and increase the bailout until the edge of the open area expanded to cut straight through the center of the midget. It took a few trials to find the exact bailout, but the image is a very fast one, so this was no problem. Once I had fixed the bailout, it was only a matter of finding a reasonable color palette. Not counting the set-up time, the attached parameter file runs in 1-1/2 minutes. Not counting the time required to go on-line, retrieving the GIF image will take about the same time. Those who decide to download the GIF image will find it waiting at: The fractal weather today was quite chilly, with brisk winds, heavy clouds, occasional spits of rain, and a temperature of 52F 11C. These conditions limited the fractal cats to only a few minutes outdoors, but they didn't complain too much, and were actually rather refined cats most of the day. I've been thinking much (a rare event) about the fourth dimension and the hypersphere lately, and have decided that the best place to start is at the surface. The surface (or hypersurface) of a hypersphere is a three-dimensional space of constant positive curvature. Limited portions of this curved space appear exactly like our familiar 'real' space, though as the range of vision grows larger, strange differences appear. One of the most intuitive features of the 'real' world is that the farther away an object lies, the smaller it appears, until it finally disappears altogether. To our everyday minds, it must be this way. Common sense dictates that it could not possibly be any other way. If our space were the surface of a hypersphere however, common sense would lead to a wrong conclusion. An object would first appear smaller with increasing distance in the expected manner, but then, once it passed beyond a distance of one quadrant (90 degrees), it would begin to appear larger, until it reached the antipode (opposite point) of the hypersphere, when the object would appear horrendously magnified, surrounding us in all directions. Of course, the motion of objects, the finite speed of light, and the ripples in space caused by the presence of mass would prevent the ultimate view from being the back of our own head. But the apparent increase in size with increasing distance should easily be observable. To my knowledge, we have not observed this effect in our universe. This may be because we cannot observe a large enough volume of space, but most likely it is because our universe is not shaped like the hypersurface of a hypersphere, but rather more like a pseudosphere. But a pseudosphere is a different animal entirely, which makes a different story. We'll not get into that story -- at least not yet. In the next FOTD I'll lay out the latitude, longitude and altitude grid on the hypersurface of a hypersphere. Until then, when we'll all have our bearings, take care, and don't get lost. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
MidgetInTheWoods { ; time=0:01:31.61--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=+1.81788303560401100/+0.010023098963900\
95/3360.9/1/80.0000000001628422/-1.592690701546217\
72e-010 params=1/-1.5/2/-15/0/245400000 float=y
maxiter=275 inside=0 logmap=20
colors=000dSWh`XifVilTjrRjxPdrM_lJVfHP`EKVCFP9AJ7I\
J6QJ5YJ4jJ3rJ3zP7mUAgZDedHciKbnNWgOQaPKWQJTRIRSHPT\
HNUGLVFJWEHXEFY7EX0EW5CXABYF9ZK8_P6`U5aY4bV6cS7dQ9\
dRAeSDeTGfUJgVMgWPhXShYVgZYf_Ze`_daacbbbccadd`ee_f\
cZgaYiZXkXWmUVoSVqPVsMWuKXwHYyFZzC_zA`zCazDazEazGa\
zHazIbzKbzLbzMbzObzPczQczSczTczUczVczXezYfzZgz_hza\
izbjzclzdmzfnzgozhpziqzjrzkkzldzmbzn`znZzoXzoVzpUz\
pSzqQzrOzrMzsLzsJztHztFzuDzuCzvDzvEzwFzwGzxHzxIzyJ\
zyKzzLzzMzzNzzOzzPzzPzyNzwMzuLzrKzpJznHzlGziFzgEze\
DzcCzbBzaAz`Az_9zZ9zY8zY8zX7zW7zV6zU6zT5zT5zcJzmXz\
kWziWzgWzfVzdVzbVzaUz_UzYUzXTzVTzTTzSSzQSzOSzNSzKW\
zI_zGczEgzBkz9oz7sz5vzArzFnzKjzPfzUbzYZzbVzgRzlNzq\
JzoFzlAziJzfSzc`z`izYqz`kzcfzfazhXzkSznNzqIzoDzmKz\
hQzcXzZbzUizPozNpzKqzIrzGszDtzBuz9vzJmzTezaYzXUzTR\
zPOzLKzHHzDEz9BzCEzEGzGIzIKzKMzMOzOQzQSzSUzUWzWYzY\
_z_bzadzcfzehzgjzilzknzmp
}
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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