Monday, February 25, 2013

Make Newton fractal with python

A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension , see Fractal wikipedia.

Today I used my mind and also my python skills to make one fractal image.

I use Newton's method to make all points and PIL python module to save the result.

Let's see the source code and comments.

from PIL import Image
#size of image
imgx = 600
imgy = 400
#make image buffer
image = Image.new("RGB", (imgx, imgy))

# area of fractal
xa = -2.0
xb = 2.0
ya = -2.0
yb = 2.0

#define constants
max_iterations = 10 # max iterations allowed
step_derivat = 0.002e-1 # step size for numerical derivative
error = 5e-19 # max error allowed

# function will generate the newton fractal
def f(z): return z * z  + complex(-0.31,0.031)

# draw derivate fractal for each y and x
for y in range(imgy):
zy = y * (yb - ya)/(imgy - 1) + ya
for x in range(imgx):
zx = x * (xb - xa)/(imgx - 1) + xa
z = complex(zx, zy)
for i in range(max_iterations):
# make complex numerical derivative
dz = (f(z + complex(step_derivat, step_derivat)) - f(z))/complex(step_derivat,step_derivat)
# Newton iteration see wikipedia
z0 = z - f(z)/dz
# stop to the error
if abs(z0 - z) < error:
break
z = z0
#I use modulo operation expression to do RGB colors of the pixels
image.putpixel((x, y), (i % 8 * 16, i % 4 * 32,i % 2 * 64))

#save the result
image.save("fractal.png", "PNG")

This is the final result of Newton fractal image: 