
A Barnsley fern is a fractal named after British mathematician Michael Barnsley and can be created using an iterated function system (IFS).



Task

Create this fractal fern, using the following transformations:

 ƒ1   (chosen 1% of the time)
        xn + 1 = 0
        yn + 1 = 0.16 yn

 ƒ2   (chosen 85% of the time)
        xn + 1 = 0.85 xn + 0.04 yn
        yn + 1 = −0.04 xn + 0.85 yn + 1.6

 ƒ3   (chosen 7% of the time)
        xn + 1 = 0.2 xn − 0.26 yn
        yn + 1 = 0.23 xn + 0.22 yn + 1.6

 ƒ4   (chosen 7% of the time)
        xn + 1 = −0.15 xn + 0.28 yn
        yn + 1 = 0.26 xn + 0.24 yn + 0.44.

Starting position: x = 0, y = 0




 
# Put this into a new file 'fern.gmic' and invoke it from the command line, like this:
# $ gmic fern.gmic -barnsley_fern
 
barnsley_fern :
  1024,2048
  -skip {"
      f1 = [ 0,0,0,0.16 ];           g1 = [ 0,0 ];
      f2 = [ 0.2,-0.26,0.23,0.22 ];  g2 = [ 0,1.6 ];
      f3 = [ -0.15,0.28,0.26,0.24 ]; g3 = [ 0,0.44 ];
      f4 = [ 0.85,0.04,-0.04,0.85 ]; g4 = [ 0,1.6 ];
      xy = [ 0,0 ];
      for (n = 0, n<2e6, ++n,
        r = u(100);
        xy = r<=1?((f1**xy)+=g1):
             r<=8?((f2**xy)+=g2):
             r<=15?((f3**xy)+=g3):
                   ((f4**xy)+=g4);
        uv = xy*200 + [ 480,0 ];
        uv[1] = h - uv[1];
        I(uv) = 0.7*I(uv) + 0.3*255;
      )"}
  -r 40%,40%,1,1,2
 