F.O.T.N. (Fractal of the Night) 17 Jan 1998 (LostHead)

Tonight's fractal, I still create fractals at night, is an MSet, the corresponding map of the Julia sets computed from a formula by Michael G. Wareman. This time I tried putting in large values for two parameters, hoping this would give me something like the normal Mandelbrot set.

z = p1*(z2 + c)2/((z2 + c) + p2) + c

Making p1 and p2 both the same large value results in their cancellation giving us

z = (z2 + c)2 + c

just two normal MSet iterations. At first glance, that is what I got. But closer views showed some discontinuities near -2 (scaled to the normal MSet) and the midget at -1.75 slipped its filaments toward -2. All rather dull. The rest looked rather like the normal MSet. Then I looked closely into the region above the 3 ball near c=0+i. Now here was some foul play, in fact a crime! A small midget, tonight's picture, was lying there beheaded!
LOSTHEAD.GIF

Figure 1. Lost Head

This called for further investigation. Now here is where a good map comes in handy. And that is what a Mandelbrot set is. A map of the Julia sets. In the Julia sets of this image, I found two clues. One, what looked like a blunt object, possibly the murder weapon.

A_WEAPON.GIF

Figure 2. A Weapon

And then, only for a second, I got a glimpse of the villain.

GNARLY.GIF

See Figure 3. Gnarly Beast

From this example, we can see how useful it is, when exploring any Julia sets, to have the corresponding Mandelbrot set ready. Enjoy your explorations.

You can see these images by running Fractint or setting your browser to The Fractal of the Night.
Jay

For the formula and par files, copy the lowing text to an ASCII file. Put PAR as its name extension. Example, Fotn17.par. 

LostHead           { ; (c) Jay Hill, 1998
          ; with large p2 and p3 the MSet is almost normal but
          ; for disconnectivity near -2 and -1.75's filaments
          ; BUT this must be a crime, it lost its head
  reset=1960 type=formula formulafile=fgz.frm formulaname=fgz-mand
  center-mag=-0.1100183/0.9224398/218
  params=100/0/100/0/4.1/0 float=y maxiter=2560 inside=86
  colors=000<14>h9g<15>000<14>S9vUAzUAw<9>N3SN3OM2LM2HL1EK0A<14>k91mA0mA0<\
  30>UA0UA1UA3<28>bJucKwcKw<61>lbymczmczmdzmdz<28>www<13>111000000
  savename=LostHead
  }

A_Weapon         { ; (c) Jay Hill, 1998
          ; The weapon used in the LOSTHEAD crime? Must be!
  reset=1960 type=formula formulafile=fgz.frm formulaname=fgz-julia
  center-mag=0.000270701/-0.000386772/13.36898/1/4.999
  params=100/0/100/0/-0.107886/0.921693 float=y maxiter=2560 inside=0
  decomp=32
  colors=000`55<5>k91mA0mA0<30>UA0UA1UA3<26>aIqbJsbJucKwcKw<59>laylbylbymc\
  zmcz<14>qmyrmyrnxsnxsox<11>www<13>111000000<14>h9g<15>000<14>S9vUAzUAw<7\
  >O4YO4VN3SN3OM2LM2HL1EK0A<7>Z55
  savename=A_Weapon
  }

Gnarly_Beast       { ; (c) Jay Hill, 1998
          ; Once you've glimpsed a beast like this, you don't soon
          ; forget.  That is why you can't sleep at night! Too many 
          ; fractal villains!!!
  reset=1960 type=formula formulafile=fgz.frm formulaname=fgz-julia
  passes=1 center-mag=0/0/18/1/10
  params=100/0/100/0/-0.1100183/0.9224398 float=y maxiter=270 inside=0
  colors=kO8000<28>_k0am0al0<35>Z7QY5RY5S<7>V9aUAcUAcUAc<44>BA1AA0AA0AA0<1\
  22>220111000000
  }

frm:FGZ-Julia   { ; (c) Jay Hill, 1998
  ; generalization of formula by Michael G. Wareman
  ; p3 is focus of Julia set
  z=pixel, c=p3:
  z1=z*z + c;
  z = p1*z1*z1/(z1 + p2) + c;
  |z| <= 64
  }

frm:FGZ-Mand  { ; (c) Jay Hill, 1998
  IF( |p3| == 0) 
    p3 = 16
  ENDIF
  c=pixel, z=sqrt(-c), bailout = real(p3):
  z1=z*z + c;
  z = p1*z1*z1/(z1 + p2) + c;
  |z| <= bailout
}

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