Fractal of the Day
by Jim Muth
FOTD -- December 02, 1998
Fractal visionaries:
The weather today was perfect. (Except for Dr. J, who seems to have exploded.) The temperatureof 64F 18C made the fractal hunting a pleasure. In fact, it was the easiest search I have ever done, because I found today's image two days ago.
The actual scene is in an Oblate plane of the Z^7+C Julibrot. I have named the picture "Heptic Oblate Fractal" because it is an order-7 fractal sliced in the Oblate direction. (Is the word for the seventh degree heptic or septic?) Of course, the extra brilliant colors show that I gave the image a bit of a boost in Photoshop 5.0.
The shape of Julia midgets is determined by which part of the Mandelbrot set lies beneath them. In the Mandelbrot direction, Julia midgets are common in the perturbed planes. These mini Julia sets are almost always found covering a Mandelbrot midget, and the shape of the J-midget is determined by what part of the underlying M-set it overlies. The underlying Mandelbrot midget can be revealed by keeping the C-coordinates unchanged and setting initial Z to 0,0.
In the Julia planes, the underlying Mandelbrot set has enlarged to infinity. Therefore all parts of the J-image overlie the same part of an infinite M-set. As a result, Julia sets are self-similar in all their parts.
However, in the four odd planes and the rotations between them, the situation is more complicated. In this case, the underlying Mandelbrot material has been stretched to infinity in only one direction, and resembles an array of parallel bands. But the overlying Julia sets still obey the same rule. They are shaped to correspond to the part of the M-set that lies beneath them, even though that M-set has been stretched to infinity in one direction. As a result, the midgets in the odd planes are arrayed in parallel bands of similar character.
Today's image shows a Julia set corresponding to the edge of the underlying Z^7+C M-set. In fact, the actual edge is visible, cutting in grossly distorted form through the center of the Julia midget. Even though the M-edge is an infinite straight line, the overlying J-sets all along this line have the shape of a Julia set associated with the edge of a M-set. Notice the smaller midget in the upper left corner. It lies over the band of M-material and is therefore filled in. At the same time, the midget in the upper right area lies over a break between Mandelbrot bands, and is therefore open.
You know, if I keep up this kind of stuff, I might even find out what I'm writing about. But at any rate, the GIF file has been posted in all its artificial glory to:alt.binaries.pictures.fractalsand to Paul Lee's web site at:http://home.att.net/~Paul.N.Lee/FotD/FotD.html
Tomorrow is near, and to celebrate the non-occasion, I will return with a scene in the Z^4+C Julibrot. To be overwhelmed with awe, check in tomorrow around this same time. Until then, take care, and be aware that fractals are magic.
Jim Muth
jamth@mindspring.com
START FORMULA================================================
multirot07-XY-ZW {; draws 6 planes and many rotations
;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J
;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J
a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994,
z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p2,
c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p3:
z=z*(sqr(z)*sqr(sqr(z)))+c,
|z| <= 36 }
END FORMULA==================================================
START PARAMETER FILE=========================================
HepticOblateMidget { ; 75min on a 486-100mhz, 640x480
reset=1960 type=formula formulafile=multmult.frm
formulaname=multirot07-XY-ZW function=ident/flip
passes=1 center-mag=+0.40569818482113700/+0.6636445\
7177911840/1.887752e+009/0.0834/178.607/-88.034
params=0/90/0/0/0.31925/0 float=y maxiter=10000
bailout=25 inside=0 logmap=250 symmetry=none
periodicity=10 colors=000lUklUinUinUipUgpUgqUgqUesUes\
UepUdoUbnU`mUZjUZiUXgUVeUTdUT`URZUPXUNVUNRULPUJNUHLUH\
JUGGUEEUCCUCAPA8P84P62P60P40P20P0<4>0P00K0<9>0K00F0\
<5>0F0000000200200420620840A60A60C80E80GA0GA0HC0JC0LE\
0NG0NG0PH0RH0TJ0TJ0VL0XL0ZN0ZN0VV8TbVRiqPqzRszRszTszT\
uzVuzVuzVvzXvzXvzZxzZxzZxz`zz`zxbzxbzvbzvdzudzuezsezs\
ezqgzogzoizmizmkzkkzkkzimzimzgozgozeozeqzdqzdszbszbsz\
`uz`uzZvzZvzXvzXuzZuzZuz`uz`uz`uzbuzbuzbuzduzduzdszes\
zeszeszgszgszgsziszisziszkszkqzkqzmqzmqzmqzoqzoqzoqzq\
qzqqzqqzsozsozsozuoxuoxuoxvoxvovvovxovxovxmsumqqmomkm\
ikigkgdie`idXibTgZRgXNgVJeTGeRCeNAdL6dJ2dH0bG0bC0bA0`\
80`60`40gE0oL0uV0zb0zi0ze0<20>z20z00z00<6>z00z0cz0cn0\
_j0cj2gdAneAqgAsg8ui8vi6zk6zk4zm4zm2zo2z
}
END PARAMETER FILE===========================================
START 19.6 PARAMETER-FORMULA FILE============================
HepticOblateMidget { ; 75min on a 486-100mhz, 640x480
reset=1960 type=formula formulafile=multmult.frm
formulaname=multirot07-XY-ZW function=ident/flip
passes=1 center-mag=+0.40569818482113700/+0.6636445\
7177911840/1.887752e+009/0.0834/178.607/-88.034
params=0/90/0/0/0.31925/0 float=y maxiter=10000
bailout=25 inside=0 logmap=250 symmetry=none
periodicity=10 colors=000lUklUinUinUipUgpUgqUgqUesUes\
UepUdoUbnU`mUZjUZiUXgUVeUTdUT`URZUPXUNVUNRULPUJNUHLUH\
JUGGUEEUCCUCAPA8P84P62P60P40P20P0<4>0P00K0<9>0K00F0\
<5>0F0000000200200420620840A60A60C80E80GA0GA0HC0JC0LE\
0NG0NG0PH0RH0TJ0TJ0VL0XL0ZN0ZN0VV8TbVRiqPqzRszRszTszT\
uzVuzVuzVvzXvzXvzZxzZxzZxz`zz`zxbzxbzvbzvdzudzuezsezs\
ezqgzogzoizmizmkzkkzkkzimzimzgozgozeozeqzdqzdszbszbsz\
`uz`uzZvzZvzXvzXuzZuzZuz`uz`uz`uzbuzbuzbuzduzduzdszes\
zeszeszgszgszgsziszisziszkszkqzkqzmqzmqzmqzoqzoqzoqzq\
qzqqzqqzsozsozsozuoxuoxuoxvoxvovvovxovxovxmsumqqmomkm\
ikigkgdie`idXibTgZRgXNgVJeTGeRCeNAdL6dJ2dH0bG0bC0bA0`\
80`60`40gE0oL0uV0zb0zi0ze0<20>z20z00z00<6>z00z0cz0cn0\
_j0cj2gdAneAqgAsg8ui8vi6zk6zk4zm4zm2zo2z
}
frm:multirot07-XY-ZW {; draws 6 planes and many rotations
;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J
;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J
a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994,
z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p2,
c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p3:
z=z*(sqr(z)*sqr(sqr(z)))+c,
|z| <= 36 }
END 19.6 PARAMETER-FORMULA FILE==============================
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