Fractal of the Day (FotD) by Jim Muth

FOTD -- May 28, 1997

Fractal visionaries:

No other fractal is quite like the Mandelbrot set. No other fractal gives so great a variety from so simple a formula. But the Mandelbrot set is but one in an endless series of similar fractals. There is no reason one must stop at Z²+C; the formula works just as well with Z³+C, Z⁴+C, or Z^n+C.

However the higher order mandeloids soon degenerate into a monotonous circle with n-1 identical bays. Indeed, when n is greater than 12, the sets are barely distinguishable from one another, and the midgets, which grow in size and number until they nearly obliterate all else, soon become uninteresting off-center circles. The only really interesting higher order mandeloids are the Z³, the Z⁴, and to some extent the Z⁵.

Today's fractal, "All Nine", is a symmetrical order 3 midget that I picked in honor of the Z³ mandeloid, which gets far less attention than it deserves. The midget is located on the negative x-axis of its parent fractal. I realize that the order 3 mandeloid doesn't normally have midgets on its negative tail, in fact it doesn't have a negative tail at all. But I cheated a bit by writing a formula that gives it one, complete with midgets.

The little two-headed midget sits surrounded by its pattern, glowing like a vision of the holy grail, with its arms radiating around it in ascending powers of three, rather than in powers of two. Thus instead of 4, 8, 16, we have 9, 27, 81. BTW, this same increasing-powers-of-the-exponent sequence continues into the higher orders, though the features soon become too tiny to be distinguished. The sequence also prevails in the fractional order mandeloids, which is why those figures must be filled with such annoying discontinuities.

A finished 640x480x256 image of "All Nine" has been posted to a.b.p.f. and a.f.p.

Jim Muth
jamth@mindspring.com

START FORMULA FILE================================================ Mytest02 {; Jim Muth z=c=pixel: z=((z*(z+p1))^p2)^p3+c, |z| < 100 } END FORMULA FILE================================================== START PARAMETER FILE============================================== All_Nine { ; 10 minutes at 100mhz, 640x480 reset=1960 type=formula formulafile=jim.frm formulaname=Mytest02 passes=t float=y maxiter=1500 bailout=100 inside=0 periodicity=10 center-mag=-1.75424934433985100/+0.00000000000000001/2.054466e+011 params=1/0/1/0/3/0 logmap=yes symmetry=xaxis colors=000`R`<5>CcrswH<7>ga`<3>3mG<6>qBj<7>v`JWUE<4>H76<2>KGXLK\ fMMd<5>ScOTfLSfQ<5>Kcs<9>6MW4KT5OX<5>BqvCftDVrFJo<7>57PCFPJNOcc\ G<25>_PF_PF_OG_OFAdk<7>F`eG`dG`cG`b<5>HaUIbSKbT<13>ueh<5>JaSC0P\ <6>L0IM0GP0FR0EQ0D<3>W08X7G<4>a2u<8>xuq<2>jUE<4>01Q<8>H1HJ1GI2I\ <5>CC_AEb7N`3Wi7eiCer<3>UeoZexcez<2>qez<3>kUz<4>ccza`za_zbczbbz\ cazdezddzeczegzffzgizaozWvzQzzZzz } END PARAMETER FILE================================================ START PARAMETER-FORMULA FILE FOR 19.6============================= All_Nine { ; 10 minutes at 100mhz, 640x480 reset=1960 type=formula formulafile=jim.frm formulaname=Mytest02 passes=t float=y maxiter=1500 bailout=100 inside=0 periodicity=10 center-mag=-1.75424934433985100/+0.00000000000000001/2.054466e+011 params=1/0/1/0/3/0 logmap=yes symmetry=xaxis colors=000`R`<5>CcrswH<7>ga`<3>3mG<6>qBj<7>v`JWUE<4>H76<2>KGXLK\ fMMd<5>ScOTfLSfQ<5>Kcs<9>6MW4KT5OX<5>BqvCftDVrFJo<7>57PCFPJNOcc\ G<25>_PF_PF_OG_OFAdk<7>F`eG`dG`cG`b<5>HaUIbSKbT<13>ueh<5>JaSC0P\ <6>L0IM0GP0FR0EQ0D<3>W08X7G<4>a2u<8>xuq<2>jUE<4>01Q<8>H1HJ1GI2I\ <5>CC_AEb7N`3Wi7eiCer<3>UeoZexcez<2>qez<3>kUz<4>ccza`za_zbczbbz\ cazdezddzeczegzffzgizaozWvzQzzZzz } frm:Mytest02 {; Jim Muth z=c=pixel: z=((z*(z+p1))^p2)^p3+c, |z| < 100 } END PARAMETER-FORMULA FILE FOR 19.6===============================