Interactive Mandelbrot Set¶

Mandelbrot Set is a fractal, which is self-similar. If you zoom in on a fractal object it will look similar, (possibly rotated), or exactly like the original shape.

There are many other known fractals .

In [3]:
from numpy import * # because arrays are defined in numpy
from numba import njit  # This is the new line with numba
from numba import prange

@njit   # this is an alias for @jit(nopython=True)
def Mand(z0, max_steps):
    z = 0j  # no need to specify type. 
    # To initialize to complex number, just assign 0j==i*0
    for itr in range(max_steps):
        if abs(z)>2:
            return itr
        z = z*z + z0
    return max_steps

@njit(parallel=True)
def Mandelbrot3(data, ext, max_steps):
    """
    ext[4]    -- array of 4 values [min_x,max_x,min_y,max_y]
    Nxy       -- int number of points in x and y direction
    max_steps -- how many steps we will try at most before we conclude the point is in the set
    """
    Nx,Ny = shape(data) # 2D array should be already allocated we get its size
    for i in range(Nx):
        for j in prange(Ny):    # note that we used prange instead of range.
                                # this switches off parallelization of this loop, so that
                                # only the outside loop over i is parallelized.
            x = ext[0] + (ext[1]-ext[0])*i/(Nx-1.)
            y = ext[2] + (ext[3]-ext[2])*j/(Ny-1.)
            # creating complex number of the fly
            data[i,j] = Mand(x + y*1j, max_steps)  
# data now contains integers. 
# MandelbrotSet has value 1000, and points not in the set have value <1000.
In [6]:
%matplotlib
import matplotlib.pyplot as plt
import matplotlib.cm as cm
#from pylab import *    # plotting library
# don't use "%matplotlib inline"

def ax_update(ax):  # actual plotting routine
    ax.set_autoscale_on(False) # Otherwise, infinite loop
    # Get the range for the new area
    xstart, ystart, xdelta, ydelta = ax.viewLim.bounds
    xend = xstart + xdelta
    yend = ystart + ydelta
    ext=array([xstart,xend,ystart,yend])
    Mandelbrot3(data, ext, 1000) # actually producing new fractal
    
    # Update the image object with our new data and extent
    im = ax.images[-1]  # take the latest object
    im.set_data(-log(data.T))   # update it with new data
    im.set_extent(ext)           # change the extent
    ax.figure.canvas.draw_idle() # finally redraw


data = zeros((1000,1000))
ext=[-2,1,-1,1]
Mandelbrot3(data, array(ext), 1000)
fig,ax = plt.subplots(1,1)
ax.imshow(-log(data.T), extent=ext, aspect='equal',origin='lower',cmap=cm.hot)
    
ax.callbacks.connect('xlim_changed', ax_update)
ax.callbacks.connect('ylim_changed', ax_update)
plt.show()

Using matplotlib backend: MacOSX
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