Don't fractal patterns occur everywhere, even in behavioral patterns?

What types of behavior can a relatively simple process create? Through the use of iteration, and time, Alan Turing and Alonzo Church posited that all every calculable function is a computable function. How does this relate to complex, infinite ideas? For example, can a fractal be proven to be computable?
Animation allows a pragmatic proof that a very simple computation process can recreate an extremely complicated and infinite idea.
This week I wanted to share a way to visualize many different 'snapshots' of a model so that through animation more complex patterns could be communicated. What you saw above was an example of a Sierpinski triangle that I created through such a process. I'll tell you how I did it in a bit. But first, let's discuss fractals.

Fractals are objects that represent figures that exist in fractional dimensions. Fractals are known to be be infinitely self-representative. Simply put, a snapshot of a fractal will still resemble the full fractal. Such a figure-ground gestalt could prove useful when it comes to modeling behaviors.

Useful? You may be wondering what practical use a fractal could have. Beyond its intriguing aesthetic qualities, fractals and similar processes can be used to explain and simplify naturally occurring events. For example, Blaise Pascal found a simple summation series of numbers that, if arranged in a triangular grid, recreates this fractal. Shading in the odd number in Pasca's arrangement re-creates the Sierpinski triangle! Don't believe me? Take a look for yourself.

As promised, I'll explain how the ideas of Church, Turing, Pascal, and Sierpinski to re-create such a fractal. Just follow these steps (Credit to Andrew Ho for this simplification):

  1. Find three points on a plane.
  2. Select a point within the triangle at random.
  3. Randomly (or pseudo-randomly as the case may be) select one of three vertices.
  4. Calculate mid-point of line between first point and this vertex.
  5. Plot that point.
  6. Repeat!

I repeated this process many times (5000 in this case) and took a snapshoot every 40 iterations. After turning these into an animation, what you see below is the result. You can find the full-size version of this animation here.

If it looks granular, see the full size image.

Join the conversation:

How could I use an animation such as this to illustrate other ideas? How could I use fractals to predict or model other ideas?