F.O.T.N. (Fractal of the Night) 21 Jan 1998 (Starfish)

And now presenting a first for Dr. J, the mad scientist. Tonight, shocked by what he saw in his fractal mirror last night (Big Fatso) he decided to go for an evening walk down to the beach. He soon arrived at the very edge of fractal land, where the solid features become fractal dust or sand. Even in the fading light, he thought for sure he saw something interesting. It was so small he had to get out his pocket Fractoscope(tm), a Fractint Model 19.6, to see it. Carefully picking it up, he took it back to the lab for a better look. Ah, a baby starfish! He knew they were small, but this one was less than a micron across. You can see its picture, shown as he saw it, by dialing in the proper settings on your fractoscope.

Figure 1. Starfish

The formula is Jim Muth's MandelbrotMix which iterates

g(z,c) = a z^b + d z^f + c

Jim was wondering about why certain forms show small MSet midgets (usuall cardioid and buds) while others show higher order shapes like z := z^3 +c. The effect comes about when we use the critical point to start the iteration. That is, z0 is a zero of this equation dg/dz = 0. For Muth's formula, critical points are

z0=(-a*b/f/d)^(1/(f-b))

I decided to explore a mix of z^5 and z^6. Each of which would produce 4 and 5 sided objects respectively. Mixed they produce distorted 5 sided objects when z0=0. The critical point for this example is -5/6 = -.83333333333.

Making image as a function of z0 show progressively more distortion as z0 decreases passed -5/6. I pushed it until the solid objects disappeared suddenly at about -1.42. In fact tonight's picture is of a starfish near the last small spec. The value of z0 has been pushed to -1.42237.

Here are the Fractint parameter files.

Enjoy

frm:MandelbrotMix {; Jim Muth z=p3, c=pixel, a=real(p1), b=imag(p1), d=real(p2), f=imag(p2): z=(a*(z^b))+(d*(z^f))+c, |z| <= 100 }

Starfish { ; (c) Jay Hill, 1998 ; 5 points distorted ; one of the last midgets as z0 increases passed critical ; upper part to the left - Julia like starfish reset=1960 type=formula formulafile=mix1.par formulaname=mandelbrotmix center-mag=-1.4216157031/+0.000455748689/1.098873e+008/1/-20 params=1/5/1/6/-1.42237/0 float=y maxiter=256000 inside=bof61 outside=0 colors=000wc0<17>wM0wK0wK0<89>wZNwZNw\ ZNwZNwZNwZN<12>w`Lw`Lw_L<121>U00000\ 000 savename=Starfish }