Mandelbrot Set with Python

Data can be beautiful.  Visualizing data is a worthwhile skill to acquire and it's relatively simple with Python.  Let's explore how to do some data visualization as an exercise.

The Mandelbrot set is often regarded as an example of art meeting science.  It's generated by evaluating the behavior of complex numbers and generating the results.  The end effect is a infinite and beautiful depiction of pure mathematics, an acid-trip of color and structure as you continuously zoom into the graph.

One of the best descriptions of the Mandelbrot set can be found here, I recommend you spend a few minutes to appreciate the concept before we begin graphing it with a simple Python snippet.

In about 30 lines of code we can create our own colorized visualization of the mandelbrot set.

Let's look at the source, then step into some of the details;
$ cat -n mandelbrot 
     1 #!/usr/bin/python
     2 import matplotlib.pyplot as plt;
     3 import sys;
     4
     5 def colorize(n):
     6   h="#%06x"%(int(n*2**23));
     7   return h;
     8
     9 xRange=[-2,1];
    10 yRange=[-1.5,1.5];
    11 incr=0.005;
    12
    13 x=xRange[0];
    14 while(x < xRange[1]):
    15   y=yRange[0];
    16   while(y < yRange[1]):
    17     c=x+y*1j;
    18     z=0;
    19     try:
    20       for k in range(50):
    21         z=z**2+c;
    22       if(abs(z) < 2):
    23         rgb=colorize(abs(z));
    24         plt.plot(x,y,'.',color=rgb);
    25     except:
    26       pass;
    27     y += incr;
    28   x += incr;
    29
    30 plt.xlim(xRange[0],xRange[1]);
    31 plt.ylim(yRange[0],yRange[1]);
    32 plt.savefig(sys.argv[1]);

Let's look over the non-Mandelbrot stuff first.  We're using the matplotlib library for our simple plotting example, line 9-10 we define our plot x and y ranges and enforce them in lines 30-31.  Finally in line 31 we save the plot to a figure rather than display to the screen.  We pass in the figure filename as a command line argument so ./mandelbrot foo.png would generate a foo.png file.  Lines 23-24 calculates a color for the pixel and plots it at (x,y).  Without going into details, the nested loop (lines 13-28) steps through the floating point 2D range defined by the x and y ranges, stepping by 0.005 (defined in line 11).  Each iteration, we selectively plot, or don't plot, a colorized pixel.  We plot a pixel if it's position is part of the Mandelbrot set.

The rest of the details are specific to calculating the Mandelbrot set.  In particular, lines 17,20-26.  The referenced video explores how and why this is done, but let's revisit some of the particulars.

Line 17 consists of the assignment of the complex number for each (x,y) position in the range.  Note 1j is the Python representation of complex number i.   This assignment was covered in the video, specifically, but easily overlooked;

The inclusion in the mandelbrot set for this position is characterized by how this complex number behaves under the influence of iteration of the function starting at 0.  This means, z starts as 0 and we repeatedly inject the f(z)=z^2+c in our loop, the end result will either blow up implying it's not in the mandelbrot set, or it'll remain bounded (e.g. <= 2) which means it's part of the set.  Lastly, we use the magnitude of the result to determine the color of the pixel.  This is optional, but it adds a level of beauty. We can represent the looping of (x,y) range and determination of it's inclusion/exclusion in the mandelbrot set by the clip from the reference video;

The end result is our visualization;

No go do something cool!