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Title: Area of Julia Set
Post by: hgjf2 on June 20, 2013, 10:02:56 AM

Yet , neither scientist could find a formula for area of JULIA SET , just by formula {z|P(P(P...P(z)...))<>infinity, P(z) = z*z+c}.
:snore: :hrmm:

Title: Re: Area of Julia Set
Post by: hgjf2 on June 20, 2013, 10:09:04 AM

Quote from: hgjf2 on June 20, 2013, 10:02:56 AM

Yet , neither scientist could find a formula for area of JULIA SET , just by formula {z|P(P(P...P(z)...))<>infinity, P(z) = z*z+c}.
:snore: :hrmm:

But I surprised that for c=-1 from formula {z|P(P(P...P(z)...))<>infinity, P(z) = z*z+c}, area of this Julia Set seem that give 1,4142135623... , and I know is equal with sqrt2.
:surprised: :surf:
I knew after I used VISUAL BASIC and C# for calculate this area with MONTE CARLO METHOD and filling space and comparing method because I don't know neither math formula for JULIA SET quadrating and calculate area. But I don't stand how the area have given sqrt2 for P(z) = z*z-1.

Title: Re: Area of Julia Set
Post by: Pauldelbrot on June 20, 2013, 11:06:09 AM

This colors the Mandelbrot set interior by the areas of Julia sets. The UF gradient is mapped to the range of areas from 0 to pi (the area of the Julia set at (0,0), which seems to be close to the maximum attained).

Code:

JuliaArea {
final:
  int ct = 0
  int p = 0
  int seed = 385472
  WHILE (p < @julpoints)
    int i = 0
    seed = random(seed)
    float r = 2*(seed/#randomrange)
    seed = random(seed)
    complex x = r + (0,2)*(seed/#randomrange)
    WHILE (i < @juliters && |x| < 2)
      x = sqr(x)+#pixel
      i = i + 1
    ENDWHILE
    IF (|x| < 2)
      ct = ct + 1
    ENDIF
    p = p + 1
  ENDWHILE
  #index = (ct*16)/(#pi*@julpoints)
default:
  title = "Julia Area"
  param julpoints
    caption = "Number of samples"
    default = 3000
    min = 1
  endparam
  param juliters
    caption = "Julia maximum iterations"
    default = 30
    min = 1
  endparam
}

Title: Re: Area of Julia Set
Post by: matsoljare on June 20, 2013, 10:59:32 PM

Yeah, that is pretty much the same as the "Supermandelbrot" rendering that i proposed a couple of years ago, not much have been done of it though:

http://www.fractalforums.com/new-theories-and-research/supermandelbrot-and-superjulia-imaging

Title: Re: Area of Julia Set
Post by: hgjf2 on June 21, 2013, 10:43:20 AM

Nice rendering on MANDELBROT SET graphic where the color show value of area of Julia set at each point.
If making same algorithm on cubic Julia Set {z|P(P(P...P(z)...))<>infinity;P(z) = z*z*z+z*z+c}, the graphic cover two cubic Mandelbrot sets have at base formula fc(z)=z*z*z+z*z+c with starting from z=0 and z=-22/27 whick is simetrical , one is another mirrored by the ax {z=a-(1/3)i;\/a<-R}

LEGEND OF MATH SYMBOLS: \/ anything to be ; <- apartening ; <> different; R set of real numbers

Title: Re: Area of Julia Set
Post by: Pauldelbrot on June 21, 2013, 12:07:58 PM

Quote from: hgjf2 on June 21, 2013, 10:43:20 AM

Nice rendering on MANDELBROT SET graphic where the color show value of area of Julia set at each point.

Thanks.

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Fractal Math, Chaos Theory & Research => Complex Numbers => Topic started by: hgjf2 on June 20, 2013, 10:02:56 AM