# Mandelbrot Set¶

## Author Jean-François Puget¶

The code used in this notebook is discussed at length in My Christmas Gift

First, let's load useful extensions and import useful modules

In [1]:
import numpy as np
from numba import jit
from matplotlib import pyplot as plt
from matplotlib import colors
%matplotlib inline

## The Escape Method¶

We first compute Mandelbrodt set with the escape method.

In [2]:
@jit
def mandelbrot(c,maxiter):
z = c
for n in range(maxiter):
if abs(z) > 2:
return n
z = z*z + c
return 0

@jit
def mandelbrot_set(xmin,xmax,ymin,ymax,width,height,maxiter):
r1 = np.linspace(xmin, xmax, width)
r2 = np.linspace(ymin, ymax, height)
n3 = np.empty((width,height))
for i in range(width):
for j in range(height):
n3[i,j] = mandelbrot(r1[i] + 1j*r2[j],maxiter)
return (r1,r2,n3)

Using Numba makes it rather fast. Note that time is roughly proportional to the max number of iterations.

In [3]:
%timeit mandelbrot_set(-2.0,0.5,-1.25,1.25,1000,1000,80)
%timeit mandelbrot_set(-2.0,0.5,-1.25,1.25,1000,1000,800)
1 loops, best of 3: 163 ms per loop
1 loops, best of 3: 1.34 s per loop

## Display¶

Let's plot the result. We take as input the size of the image in inches, and we compute the number of pixels in each axis using the standard 72 dots per inch used for web images.

In [4]:
image_counter = 30

def save_image(fig):
global image_counter
filename = "mandelbrodt_%d.png" % image_counter
image_counter += 1
fig.savefig(filename)
In [5]:
def mandelbrot_image(xmin,xmax,ymin,ymax,width=10,height=10,maxiter=256):
dpi = 72
img_width = dpi * width
img_height = dpi * height
x,y,z = mandelbrot_set(xmin,xmax,ymin,ymax,img_width,img_height,maxiter)

fig, ax = plt.subplots(figsize=(width, height),dpi=72)
ticks = np.arange(0,img_width,3*dpi)
x_ticks = xmin + (xmax-xmin)*ticks/img_width
plt.xticks(ticks, x_ticks)
y_ticks = ymin + (ymax-ymin)*ticks/img_width
plt.yticks(ticks, y_ticks)

ax.imshow(z.T,origin='lower')

save_image(fig)

Let's see what this gives

In [6]:
mandelbrot_image(-2.0,0.5,-1.25,1.25)

Let's try different color maps.

In [7]:
def mandelbrot_image(xmin,xmax,ymin,ymax,width=10,height=10,maxiter=80,cmap='jet'):
dpi = 72
img_width = dpi * width
img_height = dpi * height
x,y,z = mandelbrot_set(xmin,xmax,ymin,ymax,img_width,img_height,maxiter)

fig, ax = plt.subplots(figsize=(width, height),dpi=72)
ticks = np.arange(0,img_width,3*dpi)
x_ticks = xmin + (xmax-xmin)*ticks/img_width
plt.xticks(ticks, x_ticks)
y_ticks = ymin + (ymax-ymin)*ticks/img_width
plt.yticks(ticks, y_ticks)
ax.set_title(cmap)

ax.imshow(z.T,cmap=cmap,origin='lower')

save_image(fig)
In [8]:
mandelbrot_image(-2.0,0.5,-1.25,1.25,cmap='hot')

That's a bit dark, because most of the pixels have a low value.

In [9]:
r1,r2,z = mandelbrot_set(-2.0,0.5,-1.25,1.25,100,100,100)
plt.hist(z)
plt.show()

We can use a power norm to spread the values more evenly.

In [10]:
z3 = z ** 0.3
plt.hist(z3)
plt.show()
In [11]:
def mandelbrot_image(xmin,xmax,ymin,ymax,width=10,height=10,maxiter=80,cmap='jet'):
dpi = 72
img_width = dpi * width
img_height = dpi * height
x,y,z = mandelbrot_set(xmin,xmax,ymin,ymax,img_width,img_height,maxiter)
z = z ** 0.3

fig, ax = plt.subplots(figsize=(width, height),dpi=72)
ticks = np.arange(0,img_width,3*dpi)
x_ticks = xmin + (xmax-xmin)*ticks/img_width
plt.xticks(ticks, x_ticks)
y_ticks = ymin + (ymax-ymin)*ticks/img_width
plt.yticks(ticks, y_ticks)
ax.set_title(cmap)

ax.imshow(z.T,cmap=cmap,origin='lower')
In [12]:
mandelbrot_image(-2.0,0.5,-1.25,1.25,cmap='hot')

An equivalent way of doing it is to use a normalize object

In [13]:
def mandelbrot_image(xmin,xmax,ymin,ymax,width=10,height=10,maxiter=80,cmap='jet',gamma=0.3):
dpi = 72
img_width = dpi * width
img_height = dpi * height
x,y,z = mandelbrot_set(xmin,xmax,ymin,ymax,img_width,img_height,maxiter)

fig, ax = plt.subplots(figsize=(width, height),dpi=72)
ticks = np.arange(0,img_width,3*dpi)
x_ticks = xmin + (xmax-xmin)*ticks/img_width
plt.xticks(ticks, x_ticks)
y_ticks = ymin + (ymax-ymin)*ticks/img_width
plt.yticks(ticks, y_ticks)
ax.set_title(cmap)

norm = colors.PowerNorm(gamma)
ax.imshow(z.T,cmap=cmap,origin='lower',norm=norm)

save_image(fig)
In [14]:
mandelbrot_image(-2.0,0.5,-1.25,1.25,cmap='hot')

## Antialiasing¶

This is nice but it shows one limit of this technique: the colors aren't evolving smoothly.
The followings function computes a floating point value for each point instead of an integer. Math for it can be found in my blog

In [15]:
@jit
def mandelbrot(z,maxiter,horizon,log_horizon):
c = z
for n in range(maxiter):
az = abs(z)
if az > horizon:
return n - np.log(np.log(az))/np.log(2) + log_horizon
z = z*z + c
return 0

@jit
def mandelbrot_set(xmin,xmax,ymin,ymax,width,height,maxiter):
horizon = 2.0 ** 40
log_horizon = np.log(np.log(horizon))/np.log(2)
r1 = np.linspace(xmin, xmax, width)
r2 = np.linspace(ymin, ymax, height)
n3 = np.empty((width,height))
for i in range(width):
for j in range(height):
n3[i,j] = mandelbrot(r1[i] + 1j*r2[j],maxiter,horizon, log_horizon)
return (r1,r2,n3)
In [16]:
mandelbrot_image(-2.0,0.5,-1.25,1.25,cmap='hot')

Much better! The image is antialiased

## Zooming¶

Let's zoom on the valley between the two largest black areas

In [17]:
mandelbrot_image(-0.8,-0.7,0,0.1,cmap='hot')

We see that the precision isn't great. We need to augment the number of iterations.

In [18]:
mandelbrot_image(-0.8,-0.7,0,0.1,cmap='hot',maxiter=1024)

Win.

Let's zoom further on the 'sea horses on the right' . We need to further increase the number of iteraitons to get a crisper result.

In [19]:
mandelbrot_image(-0.755,-0.745,0.06,0.07,cmap='hot',maxiter=2048)