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From: "Jay Hill" <ehill1@san.rr.com>
To: <fractint@lists.xmission.com>
Subject: (fractint) Field lines, binary and Trinary decomposition
Date: Sat, 3 Jan 1998 21:29:23 -0800
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Hi Fractintiers,

In the last month there are at least two threads, maybe 
three, on this topic mixing and folding, in the true spirit 
of chaos and fractals.  References below are to posts 
to this forum in the last few weeks.

I hope you enjoy the formulas and pars.

Jay

        Field Lines, Binary and Trinary Decomposition
    (c) Jay Hill 1998  (as if anyone would want to copy this...)

1) There is the question of how to plot binary decomposition. 
One way is as posted, matching existing text books. That is 
choosing black or white depending on whether the bailout z is 
above the real line. Fine, but the alternative, left/right of the 
imaginary axis just shifts the phase of the black-white blocks 
and has the interesting effect of somehow correctly mapping 
the real line. Indeed, the midgets and buds on the real line 
have a numbering scheme based on this method. But as 
mentioned, it breaks down for complex midgets and buds. For 
these, a different boundary is needed. And it is not the imaginary 
axis. It is at an angle depending upon which field line you are 
following. Worse still, the boundary (for coloring black or white 
based on z) is not always straight!

2) Another thread is the issue of extending binary decomposition 
to more bits. I claimed earlier that the approach Paul Derbyshire 
suggested will not work. One reason is the curved boundaries 
mentioned in 1). In the early iterations, the real line boundary is 
not a good substitute.  Paul is trying to estimate the external 
angle corresponding to c in z<-z^2+c.  The best way to get the 
external angle is iterate to bailout and measure the angle of z. 
That is,  angle = imag(log(z)).  There is an ambiguity because of 
the number of iterations used to get to bailout and the actual period 
of the field line.  Yes, every field line has a period, namely, the 
period of the component it leads to. This brings up a problem with 
binary decomposition. The lines defined by binary decomposition 
have no period. The lines (follow the radial boundaries of the 
checker pattern) seem go to chaotic points. So, to extend binary 
decomposition to 256 levels, follow z to bailout, set the color to 
256*[imag(log(z))/(2*pi)].

3) A third thread is how do we follow field lines. That is two 
problems, how to plot them and how to follow them for pan and 
zoom. We can start with what we know from 1) and 2).  Now 
we must choose a set of field lines to plot and they must 
all have the same period (leading to components having the 
same period). Otherwise we are back in the problems of binary 
decomposition, namely colors scrambling at each iteration boundary 
curve (the ones that go around the MSet - not radial).  So we can 
choose the simplest interesting case, period 2.  There are two field 
lines going out to 120 degrees and 240 degrees from the base of the 
period two bud (-0.75).  The shape is like an hyperbola. So the 
method is iterate and select color based on the three regions. One 
wrinkle, we must do two iterations in each loop to keep the pattern 
from scrambling.  So here is a frm and par to do just that and plot lines.

frm:J-decomp3 { ; Trinary decomp (c) Jay Hill 1997
                    ; One boundary leads to -.75
                    ; Another boundary leads to +i.
done = 1               ; Iteration flag.
iter = 0               ; Sylvie Gallet's suggestion
z = 0, c = pixel:      ; Mandelbrot initialization.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
IF (|z| >= 100)        ; Point exceeds bailout.
  done = -1            ; Set flag to force an exit.
  y=.5*imag(log(z))/pi
  IF (y<0) 
    y=y+1 
  ENDIF
  w=3*y
  z=w - iter - 8       ; Sylvie Gallet's suggestion
ENDIF
iter = iter + 1        ; Sylvie Gallet's suggestion
done >= 0              ; Continue if the flag >=0.
}

The par files are at the end.   This can be extended to the next level 
and so on......  I have chosen color banding so you can follow a few 
of the lines all the way to the component base.  
  
frm:J-decomp7 { ; 7-ary decomp (c) Jay Hill 1997
done = 1               ; Iteration flag.
iter = 0               ; Sylvie Gallet's suggestion
z = 0, c = pixel:      ; Mandelbrot initialization.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
IF (|z| >= 100)        ; Point exceeds bailout.
  done = -1            ; Set flag to force an exit.
  y=.5*imag(log(z))/pi
IF (y<0) 
y=y+1 
ENDIF
  w=7*y
  z=w - iter - 8       ; Sylvie Gallet's suggestion
ENDIF
iter = iter + 1        ; Sylvie Gallet's suggestion
done >= 0              ; Continue if the flag >=0.
}

frm:J-decomp15 { ; 15-ary decomp (c) Jay Hill 1997
done = 1               ; Iteration flag.
iter = 0               ; Sylvie Gallet's suggestion
z = 0, c = pixel:      ; Mandelbrot initialization.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
IF (|z| >= 100)        ; Point exceeds bailout.
  done = -1            ; Set flag to force an exit.
  y=.5*imag(log(z))/pi
IF (y<0) 
y=y+1 
ENDIF
  w=15*y
  z=w - iter - 8       ; Sylvie Gallet's suggestion
ENDIF
iter = iter + 1        ; Sylvie Gallet's suggestion
done >= 0              ; Continue if the flag >=0.
}

frm:J-decomp31 { ; 31-ary decomp (c) Jay Hill 1997
done = 1               ; Iteration flag.
iter = 0               ; Sylvie Gallet's suggestion
z = 0, c = pixel:      ; Mandelbrot initialization.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
IF (|z| >= 10000)      ; Point exceeds bailout.
  done = -1            ; Set flag to force an exit.
  y=.5*imag(log(z))/pi
IF (y<0) 
y=y+1 
ENDIF
  w=31*y
  z=w - iter - 8       ; Sylvie Gallet's suggestion
ENDIF
iter = iter + 1        ; Sylvie Gallet's suggestion
done >= 0              ; Continue if the flag >=0.
}

frm:J-decomp63 { ; 63-ary decomp (c) Jay Hill 1997
done = 1               ; Iteration flag.
iter = 0               ; Sylvie Gallet's suggestion
z = 0, c = pixel:      ; Mandelbrot initialization.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
z = sqr(z) + c         ; Mandelbrot calculation.
IF (|z| >= 10000)      ; Point exceeds bailout.
  done = -1            ; Set flag to force an exit.
  y=.5*imag(log(z))/pi
IF (y<0) 
y=y+1 
ENDIF
  w=63*y
  z=w - iter - 8       ; Sylvie Gallet's suggestion
ENDIF
iter = iter + 1        ; Sylvie Gallet's suggestion
done >= 0              ; Continue if the flag >=0.
}

j_decomp-03       { ; Trinary decomp (c) Jay Hill 1997
                    ; Red - Blue boundary leads to -.75
                    ; Blue - White boundary leads to +i.
  reset=1960 type=formula formulafile=j_decomp.frm
  formulaname=j-decomp3 center-mag=-0.5/0/0.5 float=y maxiter=256
  inside=0 outside=real periodicity=0
  colors=000ccc00cc00cK0wwwU10U1UUU0U1wKUAUAKU10<2>U20j30<236>0m0000000
  }

j_decomp-07       { ; 7-ary decomp (c) Jay Hill 1997
                    ; White and gray lead to period 3
  reset=1960 type=formula formulafile=j_decomp.par
  formulaname=j-decomp7 center-mag=-0.5/0/0.5 float=y maxiter=256
  inside=0 outside=real periodicity=0
  colors=000ccc00cc00cK0wwwU10U1UUU0U1wKUAUAKU10<2>U20j30<236>0m0000000
  }

j_decomp-15       { ; 15-ary decomp (c) Jay Hill 1997
  reset=1960 type=formula formulafile=j_decomp.par
  formulaname=j-decomp15 center-mag=-0.5/0/0.5 float=y maxiter=256
  inside=0 outside=real periodicity=0
  colors=0AKKKK00zK00000KAUK1AU1KKK0z0zwwwKAAAKUUKKKAKUA0j30<236>0m0000000
  }

j_decomp-31       { ; 31-ary decomp (c) Jay Hill 1997
  reset=1960 type=formula formulafile=j_decomp.par
  formulaname=j-decomp31 center-mag=-0.5/0/0.5 float=y maxiter=256
  inside=0 outside=real periodicity=0
  colors=0AKkVzMMMdVPwcSKKKcUIwwFKmDKwAhhh<4>wwwU0AAAU6DX<3>KcwKEa<3>WMkZO\
  naQqdStgUwg60<220>0m0000000
  }

j_decomp-63        { ; 63-ary decomp (c) Jay Hill 1997
  reset=1960 type=formula formulafile=n-ary.par formulaname=j-decomp63
  center-mag=-0.5/0/0.5 float=y maxiter=2560 inside=0 outside=real
  periodicity=0
  colors=0AK<7>www9CU<5>FT`ACO<14>VJwWKA<6>wM8000<22>zU0U00<188>0U0000000
  }


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