Python PyGame: 2.1 (Mandelbrot)

Python PyGame:2.1 Mandelbrot (a short aside)

In my last blog I started off with a game that draws circles of different colors. That is boring. Personally, I never tire of rewriting programs to draw fractals. Today we’ll quickly jump through a program to draw the Mandelbrot set. Drawing 2d fractals may be so 1980, but, I get a kick out of it.

What is a fractal? Simply put, a fractal is worlds within worlds. It’s a geometric shape that no matter how far you zoom into or out of still holds the original shape. It’s a fairly new term coined by Benoit Mandelbrot in 1982 in his book, “The Fractal Geometry of Nature”.

More importantly, fractals are the closest approximation we have come up with for a Mathematics of Nature. They can approximate clouds, coastlines, snowflakes and earthquakes. It’s related to the popularization of Chaos Math by Edward Lorenz and his study of weather prediction in 1961. I cannot claim understanding, but, I do find it fascinating to ponder over. More on that after the code if you’re interested.

Python & PyGame by themselves don’t just carry around the code necessary to pull this off. This gives us an excuse to play with two cool libraries: NumPy and PIL (Python Imaging Library).
http://numpy.scipy.org/
http://www.pythonware.com/products/pil/

Once again, if you’re running Windows 32 bit, life is obvious, if you’re running 64 bit, you must download them from here:
http://www.lfd.uci.edu/~gohlke/pythonlibs/

Next I happened across a version of the program written in 2006. Updated it a bit to work with the newer libraries, and the result:

Run it

You’ll see:

That’s much cooler than a bunch of circles.

Let’s quickly review what some of those NumPy functions do.
Arange – numpy.arange([start], stop[, step], dtype=None, maskna=False)
returns an evenly spaced set of values. Sp, arange(3)=[0,1,2]

Ravel – numpy.ravel(a, order=’C’)
Returns a flattened array. So a two dimensional array of [123] and [abc] is returned as [123abc]

Shape – ndarray.shape
Tuple of array dimensions.

Zeros – numpy.zeros(shape, dtype=float, order=’C’)
Return a new array of given shape and type, filled with zeros.

Here’s some further amateur discussion:
Free Will or Determinism is one of philosophy’s Coke vs. Pepsi discussions. You’ve seen it again and again through popular media: books (Ender’s Game) or movies (Matrix, Inception). In mathematics it has shown up as well. Newtonian Mechanics is pro-determinism. Then Quantum Mechanics became popular, which many felt was more aligned with Free Will. Then Chaos Math came back as a pro-determinism mathematics that can solve a lot of seemingly “random” problems in math: cloud formations, turbulence, weather patterns. The fractal is a mainstay in Chaos Math. Totally predictable and matches up with a lot of patterns in nature. Unfortunately there remain many unsolved problems in these mathematics, but it is fun for me to think about when I’m having trouble with a simple coding problem.

Here’s some more links:
This web page explains the math, theory, and how to do this in c++:
http://warp.povusers.org/Mandelbrot/

2 comments


  • Howdy are using WordPress for your site platform?
    I’m new to the blog world but I’m trying to get
    started and set up my own. Do you require any html coding expertise to make your
    own blog? Any help would be really appreciated!

    09/21/2014
    • boss

      Not at all. WordPress is made so non-web people can get the word out; whatever that word might be. Cheers.

      11/23/2014

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