Re: [Fractint] Yet another trivial...

25 Jul 2002

      While toying with the Lyapunov Mandelbrot formula recently 
posted by Morgan Owens, I noticed that the formula produces 
images virtually identical to those drawn by the Fractint 
type=hypercomplex Mandelbrot formula.  I have included two 
parameter files, one of which uses the Lyapunov Mandelbrot 
formula, while the other uses the hypercomplex Mandelbrot 
formula, yet the two files draw identical images.

Perhaps one of the math experts on the list can explain what 
is happening here. . . .

Jim Muth
jamth@mindspring.com

START PARAMETER FILE=======================================

Lyapunov           {
  reset=2002 type=formula formulafile=allinone.frm
  formulaname=LyapunovMandel
  center-mag=-1.07668/7.77156e-016/1.344086
  params=0/0/0/1/16/0 float=y maxiter=500
  inside=0 logmap=yes periodicity=0 sound=off
  colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\
  zLzzz000555888BBBEEEHHHKKKOOOSSSWWW___ccchhhmmmsss\
  zzz00zG0zV0zj0zz0zz0jz0Vz0Gz00zG0zV0zj0zz0jz0Vz0Gz\
  00z00zG0zV0zj0zz0jz0Vz0GzVVzbVzjVzrVzzVzzVrzVjzVbz\
  VVzbVzjVzrVzzVrzVjzVbzVVzVVzbVzjVzrVzzVrzVjzVbzhhz\
  lhzqhzuhzzhzzhuzhqzhlzhhzlhzqhzuhzzhuzhqzhlzhhzhhz\
  lhzqhzuhzzhuzhqzhlz00S70SE0SL0SS0SS0LS0ES07S00S70S\
  E0SL0SS0LS0ES07S00S00S70SE0SL0SS0LS0ES07SEESHESLES\
  OESSESSEOSELSEHSEESHESLESOESSEOSELSEHSEESEESHESLES\
  OESSEOSELSEHSKKSMKSOKSQKSSKSSKQSKOSKMSKKSMKSOKSQKS\
  SKQSKOSKMSKKSKKSMKSOKSQKSSKQSKOSKMS00G40G80GC0GG0G\
  G0CG08G04G00G40G80GC0GG0CG08G04G00G00G40G80GC0GG0C\
  G08G04G88GA8GC8GE8GG8GG8EG8CG8AG88GA8GC8GE8GG8EG8C\
  G8AG88G88GA8GC8GE8GG8EG8CG8AGBBGCBGDBGFBGGBGGBFGBD\
  GBCGBBGCBGDBGFBGGBFGBDGBCGBBGBBGCBGDBGFBGGBFGBDGBC\
  G000000000000000000000000
  }

Hypercomplex       {
  reset=2002 type=hypercomplex function=sqr
  center-mag=-0.5759/7.77156e-016/1.344086
  params=0/0/0/0.5 float=y maxiter=500 bailout=7
  inside=0 logmap=yes periodicity=0 sound=off
  colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\
  zLzzz000555888BBBEEEHHHKKKOOOSSSWWW___ccchhhmmmsss\
  zzz00zG0zV0zj0zz0zz0jz0Vz0Gz00zG0zV0zj0zz0jz0Vz0Gz\
  00z00zG0zV0zj0zz0jz0Vz0GzVVzbVzjVzrVzzVzzVrzVjzVbz\
  VVzbVzjVzrVzzVrzVjzVbzVVzVVzbVzjVzrVzzVrzVjzVbzhhz\
  lhzqhzuhzzhzzhuzhqzhlzhhzlhzqhzuhzzhuzhqzhlzhhzhhz\
  lhzqhzuhzzhuzhqzhlz00S70SE0SL0SS0SS0LS0ES07S00S70S\
  E0SL0SS0LS0ES07S00S00S70SE0SL0SS0LS0ES07SEESHESLES\
  OESSESSEOSELSEHSEESHESLESOESSEOSELSEHSEESEESHESLES\
  OESSEOSELSEHSKKSMKSOKSQKSSKSSKQSKOSKMSKKSMKSOKSQKS\
  SKQSKOSKMSKKSKKSMKSOKSQKSSKQSKOSKMS00G40G80GC0GG0G\
  G0CG08G04G00G40G80GC0GG0CG08G04G00G00G40G80GC0GG0C\
  G08G04G88GA8GC8GE8GG8GG8EG8CG8AG88GA8GC8GE8GG8EG8C\
  G8AG88G88GA8GC8GE8GG8EG8CG8AGBBGCBGDBGFBGGBGGBFGBD\
  GBCGBBGCBGDBGFBGGBFGBDGBCGBBGBBGCBGDBGFBGGBFGBDGBC\
  G000000000000000000000000
  }

frm:LyapunovMandel {
	narg=real(p2)
	nmag=imag(p2)
	bailout=4*(real(p3)==0)+real(p3)*(real(p3)!=0)
	z0=c0=pixel
	z1=c1=pixel+nmag*(cos(narg)+(0,1)*sin(narg))
:
	z0=z0*z0+c0
	z1=z1*z1+c1
	cabs(z0-z1)<bailout
}

END PARAMETER FILE=========================================