Making the Mandelbrot

I've taught students to draw fractals like the Koch Snowflake in Logo and Python. The Sierpinski triangle and the Dragon Curve aren't hard to figure out, either. But the Mandelbrot Set always seemed like a different animal altogether. Sure, every math textbook has pictures of the famous pear-shaped fractal, but none teach you how to create it. Students and teachers get the feeling it must be something unapproachable, requiring years of training, but, predictably enough, a computer makes it easy.

It boils down to taking a point in the complex plane and iterating it using the formula:

c is a complex number of the form a + bi, where i is the square root of -1. Z is set to 0 initially, then you square it and add c. The result becomes the new Z. Square it and add c. Repeat a bunch of times. I think it's a great exercise for Precalculus students to write a program to do this.

As Paul Lutus explained visually, most points will get really large, and fly off out of the "control radius":

the ones that stay within the control radius are in the Mandelbrot Set.

So at first the set of points that don't fly off is big:

But iteration by iteration more and more points fly off ("diverge") and the blob starts to take the shape we know and love:

The challenge is how to take points (especially ones with a real part and an imaginary part) and put the coordinates into a function over and over. Many online examples require the use of numpy and other (very useful) Python add-ons, but I wanted to do it with only "stock" Python tools.

First I had to define a function to square the complex number a bunch of times. Every programming language has a way to handle lists, and Python's way is pretty easy. I made a 2-item list of numbers: the first item is the real part of the complex number and the second item is the imaginary part. Here's how to represent the complex number 2 + 3i:

Anybody familiar with multiplying polynomials can get their noggin around this:

The only thing separating this Precalculus exercise from an Algebra one is imaginary numbers, which I've mentioned before. When the imaginary part is squared it becomes real but negative. Now squaring complex numbers is easy:

This means the square of 2 + 3i is -5 + 12i and the square of 1 - i is -2i (the real part of the last list is zero).

Now it's just a matter of calling the squareZ function then adding c back in. This next function just tells you whether the squaring-adding is making the numbers get really big. If so, the original c is not in the Mandelbrot set.

If after all your iterating the point hasn't gotten more than 2 units away from the origin it's in the Mandelbrot set. So in the above blobs that point would be colored black. That was the tricky part. What's the best way to display the results? Will the easiest way look the best?

The original output in 1978 was text, and Paul Lutas' page contains the code needed to generate it. Here's my version, which didn't all fit on the screen at once:

It's exciting just to get that far! But I knew my students would probably want something more professional looking.

In his excellent book Python for Kids, Jason Briggs showed how to use Tkinter, the default graphics module in Python, to draw shapes and create a video game. So in my program the loops go through the rows and columns of the screen, iterating each point and coloring those in the Mandelbrot Set black. In an 800 x 800 screen that means it has 640,000 points to iterate!

To set up the screen, you have to import tkinter and put this code at the top of the program:

Finally, the main loop of the program looks like this:

The program starts at (xlow, ylow) and checks the point. If it stays within the control radius after 20 iterations it creates a one-pixel black square, otherwise it doesn't do anything and moves on to the next point. Here's how it looks:

The color pictures you see online are made by coloring the points that diverge every step. This requires some changes to the code (which I'll spare you) but it makes it look pretty. You'll recognize the colored shapes as the black blobs from above:

Python is a (relatively) easy way to explore ideas like iteration that would otherwise be impossible with just pencil and paper. Try it!