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Fractal of the Day
by Jim Muth
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Fractal visionaries: I had planned on discussing in today's FOTD the ultimate question of life, and the way it relates to the realm of fractal geometry. But I just couldn't muster the proper degree of profundity that one must feel before tackling such a life-altering topic. So instead, I dredged up a picture of a baby Mandelbrot. The guilty formula is my MandelbrotMix4, which automatically calculates a critical point of the formula and sets that value as the starting point of Z. The actual equation being iterated is (4*Z^3)+(2*Z)+C. These values draw a small Mandelbrot set rotated 90 degrees, with a curious tail dropping down from the cleft of East (Elephant) Valley, which has now become a South Valley. The main Mandelbrot figure is fairly conventional, but the East Valley tail is filled with cut-off bits and pieces that demand further investigation. Only a slight magnification is needed to reveal the tell-tale patterns that lead to hidden midgets. At first glance, today's image appears to have come from a spiral in Elephant Valley itself. A quick examination, however, will show that the pattern around the midget is more delicate, with more fine, hairline detail than that which appears around the midgets in Elephant Valley. Also, the features are broken rather than connected, as they are in the valley. I gave the image a pastel-like palette of colors, which emphasizes the more delicate nature of this midget. The tendrils curving away from the central midget are themselves filled with midgets, but alas, these midgets are beyond the reach of conventional DP math. (Hint, hint . . . Fractint developers, wouldn't it be great if arbitrary precision math could be applied to formula type fractals?) Today's harmless little midget has been posted to the usual sites -- alt.fractals.pictures and alt.binaries.pictures.fractals It is also available at Paul Lee's web site at: Tomorrow, I might gather my profundity and discuss the ultimate question of life, or then again, I might blather on about something of little consequence, but whatever I do, I will have another fractal ready and waiting. Until then, take care, and the next time you see a fractal, view it with reverence, for that fractal might encode the answer to the riddle of the universe. Jim Muth jamth@mindspring.com |
START FORMULA================================================
MandelbrotMix4 {; Jim Muth
c=pixel, 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:
z=(a*(z^b))+(d*(z^f))+c,
|z| <= 100
}
END FORMULA==================================================
START PARAMETER FILE=========================================
Pink_Midget { ; 40min on a 486-100, 640x480
reset=1960 type=formula formulafile=critical.frm
formulaname=MandelbrotMix4 passes=t
center-mag=-0.00218445795124279/-0.23466738826017880/1.0\
17131e+011 params=4/3/2/1 float=y maxiter=6000 bailout=25
inside=0 logmap=123 symmetry=none periodicity=10
colors=00016727929A2CB<5>3LI3MJ4NL4OM4PN4QO4SP<4>5TV<4>P\
beUehYhl<2>lqupsxstxutyuqvuottlr<3>vgmwflxck<2>yXgzVfzUe\
<2>rUYoVVmUTjUQgTNeTL<3>VJATH8QG5OF3PE4QD5<5>ZJB_KC_LD`P\
E_TFbVFcXFeYFe_FV`F<2>VcFVcFTdEReDPfCOfB<2>Jl9Hm8Fp7<2>A\
u59w47w35w24v2KncQamVQxO9x<16>LpKFeOWV_bLia5o<11>hzq<6>D\
WV9WSEXZ<2>Qhq<7>U8YU4WT3U<14>M2FM2EN4F<2>P9FQGF<16>blFc\
nFcqF<7>Sym<18>xzk<15>Czy<7>SzxEzz0zz<2>1zz
}
END PARAMETER FILE===========================================
START 19.6 PARAMETER-FORMULA FILE============================
Pink_Midget { ; 40min on a 486-100, 640x480
reset=1960 type=formula formulafile=critical.frm
formulaname=MandelbrotMix4 passes=t
center-mag=-0.00218445795124279/-0.23466738826017880/1.0\
17131e+011 params=4/3/2/1 float=y maxiter=6000 bailout=25
inside=0 logmap=123 symmetry=none periodicity=10
colors=00016727929A2CB<5>3LI3MJ4NL4OM4PN4QO4SP<4>5TV<4>P\
beUehYhl<2>lqupsxstxutyuqvuottlr<3>vgmwflxck<2>yXgzVfzUe\
<2>rUYoVVmUTjUQgTNeTL<3>VJATH8QG5OF3PE4QD5<5>ZJB_KC_LD`P\
E_TFbVFcXFeYFe_FV`F<2>VcFVcFTdEReDPfCOfB<2>Jl9Hm8Fp7<2>A\
u59w47w35w24v2KncQamVQxO9x<16>LpKFeOWV_bLia5o<11>hzq<6>D\
WV9WSEXZ<2>Qhq<7>U8YU4WT3U<14>M2FM2EN4F<2>P9FQGF<16>blFc\
nFcqF<7>Sym<18>xzk<15>Czy<7>SzxEzz0zz<2>1zz
}
frm:MandelbrotMix4 {; Jim Muth
c=pixel, 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:
z=(a*(z^b))+(d*(z^f))+c,
|z| <= 100
}
END 19.6 PARAMETER-FORMULA FILE==============================
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times.
