Julia Sets

Fractal Madness

 

 

 

iter = 150   'number of iterations of a selected coordinate, more makes for more detail
xorg = 0.1  'starting x coordinate on the Cartesian plane (the Real portion)
yorg = 0.5608   'starting y coordinate on the Cartesian plane (the Imaginary portion)
x = xorg
y = yorg

  'inc=how far we will step the x and y coords each time

  'you can see that this is a pretty fine stepping increment, just covering a very small area
inc = 0.000000001  

  'kx and ky are constants added to the result of each iteration calculation

kx = -0.745 
ky = 0.15

  'xp and yp are screen coordinates, assuming 600 X 400 display

For xp = 1 To 600 Step 0.5
    x = x + inc
    xs = x
    y = yorg
    ys = y
    For yp = 1 To 400 Step 0.5
        For i = 1 To iter
              'with xx being the real part and yy being the imaginary part of the complex

              'number -- we are squaring the complex number and adding the Julia

              'constants after each iteration             

            xx = x * x - y * y
            yy = 2 * x * y
            x = xx + kx
            y = yy + ky

              'this is the "jump out" limit, if the result diverges
            If Abs(x) > 2500000 Or Abs(y) > 2500000 Then i = iter
        Next

          'figure the colors to plot, based on how big the iterative calculation got

        If xx <> 0 Then nc = Log(Abs(xx) ^ 14)
        cc = Abs(nc)
        If yy <> 0 Then nc = Log(Abs(yy) ^ 8)
        cd = Abs(nc)
        If xx <> 0 And yy <> 0 Then nc = Log(Abs(xx * yy) ^ 2)
        ce = Abs(nc)
        If cc > 256 Then cc = 256
        If cd > 256 Then cd = 256
        If ce > 256 Then ce = 256
          'plot the point (this PSET plot function is given here in VB format)

        Picture1.PSet (xp, yp), RGB(cc, cd, ce)
          'step the complex coordinate and do it for the next screen point

        ys = ys + inc
        y = ys
        x = xs
   Next
Next

Below is an example of the graphical output of this kind of code, given particular starting parameters (number of iterations, step size, starting x and y coordinates, k numbers, "jump out" limit). I must admit that I think this image also included a trick routine to "shade" the pixels that were on the "underside".  I will let you explore how to do that yourself...

 Back to Recreational Computing... 

 

Julia sets are mathematical functions defined by iterative squaring of x-y coordinates in the complex number plane. I have played around with these for years on my PC. Here is a piece of Basic code that will produce Julia sets: