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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: I named today's image for its basically random color palette. Except in the areas near the center, where I emphasized the midget's surroundings with a soft green glow, the entire scene is colored with one of the random palettes created with the <enter> key. The resulting fractal is vivid, though the disorganization of its colors holds its rating to no more than a 6-1/2. The name "Randomized Colors" needs no explanation. The parent fractal of today's scene has perhaps the most irregular shape that I have yet come upon, and likely will produce several more FOTD's before being discarded to the fractal archives. It is well worth an outzoom and a look. But before doing an outzoom, reset the logmap to 0. The 17-minute render time from the attached parameter file is a bit trying. The better choice is to download the GIF image from Paul's web site at: With the fractal image under control, it's time to return to the four-dimensional abstraction known as the hypersphere. Like a spherical 3-D planet, a hyperspherical 4-D planet has an equator, a great circle that lies in the plane of rotation. But unlike a 3-D planet, a hyperplanet also has a polar great circle. This polar circle marks the axis of rotation. As the hyperplanet rotates, the points of the equator rotate in the normal 3-D manner, but every point along the polar circle turns on itself, remaining fixed in position exactly like the two polar points of earth. In the case of a 3-D planet, the rotation is simple. Due to the gyroscopic effect, the axis and its orientation will remain fixed unless an outside force is applied, which will cause the axis to rotate. With a 4-D hyperplanet however, the situation is far more complicated. The simple single rotation is a special idealized case, which would almost never be realized. The complication lies in the polar circle, which, even while acting as the axis of rotation of the equator, may itself begin rotating exactly as the equator is already doing, putting the hyperplanet into an entirely new state known as a double rotation. If the two rotations are equal, every point of the hyperplanet but the center point moves in a circle, while the center point remains fixed. If the rotations are unequal, every point spirals around and moves along a circular line, in a screw motion, tracing out a path known as a surface of double revolution, which is a two-dimensional surface curved in four dimensions. This surface somewhat resembles a doughnut. This hyperdoughnut shape is also the shape of the latitude surfaces of a hyperplanet. And in the next FOTD we'll actually lay out a latitude, longitude and altitude grid on our hyperplanet. The fractal weather today (Sunday) was brilliantly sunny but with a chilly temperature of 51F 10.5C. The winds were lighter however, and this permitted the fractal cats to sleep for over an hour in the warm sun of the porch. Unfortunately, being neither a fractal nor a cat, I cannot sleep in the sun. Instead, I find myself with a day's work that needs to be done before I turn to the fantasy world of fractals and hyperspace. But despair not, for the probability is 99.975325 percent that I shall return tomorrow around this same time with another glorious fractal and a few words not quite as glorious. Until then, take care, and if you take perfect care, nothing more will be needed. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Randomized_Colors { ; time=0:16:53.15--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=ident passes=1
center-mag=-1.02460269414871400/+0.015803598960749\
27/7397888/1/37.5000008972091194/-4.40023098798036\
916e-007 params=0.439/2.466/1.265/1.372/0/0
float=y maxiter=1800 inside=0 logmap=315
colors=000PcnLXcHPUCKKXEco7vz1zl0rU0Y90CMEMYRVhcds\
olZprFrw7nz1kz0nz0pz4050as0O`09HdzhIcL0zz0iw0Uc0CI\
FKAzszzdxdPdI9ILzv7`SzH2wA0c40I00UaSCICPmXAIEziMoU\
EOC4zPOa99RzOPhzAKYFhz9Xo4KY05FAzA4h40K0zzzsix`UdH\
CItA70zn0t`0`O0F9Uz1Io05O0Enz7Xd1EIzolz4zp1tZ0aF0H\
E7r92Z40M009Aex4Rc0AI0zS0XC9oh4aX1OL097g0zV0nK0X70\
EzFHVOzMHoE9Y41Fo54a21O00900zKza5Vxz0U`0t00`00H00x\
9Y0Vz0Lo0CY02FMIz95VzUzsKx`AdH1Iz0zzwzVScA0`40M009\
ozzYgsFKRIzz5ZY0HI0AC045000rAsP1R`0EPEztY9`L4H700v\
00R0MezFUl9IX15E91r40c10P00AKAz71c0Ve0CKzzaHzV5XE7\
t41R01zzszz`peHOKzz1sp0RO0zKnwC`c5OI09rF7P417zc2dP\
0IAE1a90R40H0055IV19K017zCpi5YL0FIV0CL05C0020OcgHK\
V9MK1O7zPrxdPlSwMUSMVz9XXzYzhZgL`KCaP2cA2dz0ew0gc0\
hI4ig1kS0lCsnz`odHpIZrRPsIFtA4v25wpixwrzFZz7Fz1Vz1\
gzzzzzvzwazcHzIOzsEz`4zHczpzzMezEKz4SzzSzdHzP5zAzz\
0azxRzhHzU5zC0zzzz`czHzzt
}
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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