Fractal of the Day
by Jim Muth
FOTD -- December 01, 1998
Fractal visionaries:
It was warm again today here at Fractal Central, though not as ridiculously warm as yesterday. The temperature of 65F 18C and hazy sun made it perfect for fractal hunting, and the fractal hunting was as hot as the temperature.
As promised yesterday, I have a scene in the Z^5+C Julibrot. This picture is 90 degrees from qualifying as a Mandelbrot scene, and also 90 degrees from being a Julia scene. It is closest to being a parabolic scene, but to my knowledge such things have yet to be defined. So let's call it a picture of a Julibrot bubble and avoid the confusion.
To begin, notice that the central hole closely resembles a pinched Julia set intersecting very near the edge of its related M-set. In the Z^5 figure, the Julia midgets have five lobes, while the Mandelbrot midgets have only four lobes, and the negative Mandelbrot figures have six lobes. There is a legitimate mathematical reason for this discrepancy, but I will not bore the FOTD readers with it. (Mainly because I can't remember it.)
The important thing is that the open area at the center of the picture resembles a Julia set. This means that, even though it is not strictly a Julia set, it is composed of Julia, (low and medium iteration), material, and will therefore resemble a Julia set.
Julia material twists and turns in all directions in the four-dimensional Julibrot, while Mandelbrot material forms 4-D hyper-cylinders. This means that Julia-like shapes can appear in any orientation within the Julibrot, but Mandelbrot shapes are limited to the Mandelbrot direction, and even then mostly to the so-called "true" Mandelbrot set, which is the slice right down the center of the Julibrot.
I named today's festive fractal "Military Fractal" because the pentagonal shape around the central figure reminds me of a certain immense office building just across the river from Washington, D.C., where most of the USA military decisions are made. The finished fractal has been posted with military protocol in GIF format to:alt.binaries.pictures.fractalsand to the WWW at:http://home.att.net/~Paul.N.Lee/FotD/FotD.html
Tomorrow I'll return with a scene in the Z^7+C figure, which is surprisingly interesting when carefully examined. Until then, take care and see you soon.
Jim Muth
jamth@mindspring.com
START FORMULA================================================
multirot05-XZ-YW {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p2, c=e*imag(pixel)+p3:
z=z*(sqr(sqr(z)))+c,
|z| <= 36 }
END FORMULA==================================================
START PARAMETER FILE=========================================
Military_Fractal { ; 20min on a 486-100, 640x480
reset=1960 type=formula formulafile=multmult.frm
formulaname=multirot05-XZ-YW passes=1 center-mag=+0.\
00001581435933467/+0.00000168349702804/2.329959e+013\
/0.01237/-5.868/89.829 params=168.966/-3.873/0.77969\
1485891/0.290109303949/0.779691485891/0.290109303049
float=y maxiter=3600 bailout=25 inside=255 logmap=342
symmetry=none periodicity=10
colors=00046J86J<6>78M68N78N<2>98PA8PA8PB8PB8O<35>X8K\
Y8KX7L<25>o9Bp9Br99<15>_9aZ9cW7f<23>uY2<39>sN5sN5rO6\
<39>buBbuCavCavCawCawCbxCcxC<2>fzIgzKhzMizOjzQ<8>szft\
zZuzRvzE<2>yzUzzUzzZzzFzzFzzHzzFzz9zzAzzCzz9zz6\
<5>zzG0UZ
}
END PARAMETER FILE===========================================
START 19.6 PARAMETER-FORMULA FILE============================
Military_Fractal { ; 20min on a 486-100, 640x480
reset=1960 type=formula formulafile=multmult.frm
formulaname=multirot05-XZ-YW passes=1 center-mag=+0.\
00001581435933467/+0.00000168349702804/2.329959e+013\
/0.01237/-5.868/89.829 params=168.966/-3.873/0.77969\
1485891/0.290109303949/0.779691485891/0.290109303049
float=y maxiter=3600 bailout=25 inside=255 logmap=342
symmetry=none periodicity=10
colors=00046J86J<6>78M68N78N<2>98PA8PA8PB8PB8O<35>X8K\
Y8KX7L<25>o9Bp9Br99<15>_9aZ9cW7f<23>uY2<39>sN5sN5rO6\
<39>buBbuCavCavCawCawCbxCcxC<2>fzIgzKhzMizOjzQ<8>szft\
zZuzRvzE<2>yzUzzUzzZzzFzzFzzHzzFzz9zzAzzCzz9zz6\
<5>zzG0UZ
}
frm:multirot05-XZ-YW {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p2, c=e*imag(pixel)+p3:
z=z*(sqr(sqr(z)))+c,
|z| <= 36 }
END 19.6 PARAMETER-FORMULA FILE==============================
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