the Buddhabrot

As we have seen, the Mandelbrot set can be calulated by iterating a function, and testing whether the sequence it generates diverges or not.  The so-called Buddhabrot (named for its appearance) looks at the trajectory of the generated sequence, as the generated points hop about, diverging or not. Points that start with a \(c\) value outside the Mandelbrot set, and so produce diverging sequences, may nevertheless have points along the way that land inside the set.

The Buddhabrot takes all the points ourside the set, and plots each point in the trajectory to divergence.  This leads to a density map: some points are visited more often than others (it is conventional to plot this rotated 90 degrees, to highlight the shape):

42 million randomly chosen values of \(c\)

In addition, there is the anti-Buddhabrot, which plots the trajectories of all the \(c\) values inside the Mandelbrot set.  These do not diverge: some converge inside the set, others cycle around.

Again, the code is relatively simple, and you can calculate the Buddhabrot and anti-Buddhabrot at the same time:

import numpy  as np
import matplotlib.pyplot as plt

IMSIZE = 2048 # image width/height 
ITER = 1000

def mandelbrot(c, k=2): 
    # c = position, complex; k = power, real
    z = c
    traj = [c]
    for i in range(1, ITER): 
        z = z ** k + c 
        traj += [z]
        if abs(z) > 2: # escapes
            return traj, []
    return [], traj
    
def updateimage(img, traj):   
    for z in traj:
        xt, yt = z.real, z.imag
        ixt, iyt = int((2+xt)*IMSIZE/4), int((2-yt)*IMSIZE/4)
        # check traj still in plot area
if 0 <= ixt and ixt < IMSIZE and 0 <= iyt and iyt < IMSIZE:   
            img[ixt,iyt] += 1    

# start with value 1 because take logs later
buddha = np.ones([IMSIZE,IMSIZE]) 
abuddha = np.ones([IMSIZE,IMSIZE]) 
for i in range(IMSIZE*IMSIZE*10):
    z = np.complex(np.random.uniform()*4-2, np.random.uniform()*4-2)
    traj, traja = mandelbrot(z, k)
    updateimage(buddha,traj)
    updateimage(abuddha,traja)
                
buddha = np.square(np.log(buddha)) # to extend small numbers
abuddha = np.log(abuddha)          # to extend small numbers

plt.axis('off')
plt.imshow(buddha, cmap='cubehelix')
plt.show()    
plt.imshow(abuddha, cmap='cubehelix')
plt.show()

These plots are more are computationally expensive to produce than the plain Mandelbrot set plots: it is good to have a large number of initial points, and a long trajectory run.  There are some beautifully detailed figures on the Wikipedia page.

As before, we can iterate using different powers of \(k\), and get analogues of the Buddhabrot.

\(k = 2.5\), the "piggy-brot"

More figures and animations of the Mandelbrot set, the Julia set, and the Buddhabrot, are available on my website.