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: