Most of the fractals that I’ve written about so far – including all of the L-system fractals – are
examples of something called iterated function systems. Speaking informally, an iterated function
system is one where you have a transformation function which you apply repeatedly. Most iterated
function systems work in a contracting mode, where the function is repeatedly applied to smaller
and smaller pieces of the original set.
There’s another very interesting way of describing these fractals, and I find it very surprising
that it’s equivalent. It’s the idea of an attractor. An attractor is a dynamical system which, no matter what its starting point, will always evolve towards a particular shape. Even if you
perturb the dynamical system, up to some point, the pertubation will fade away over time, and the system
will continue to evolve toward the same target.
The classic example of this is something called the chaos game. In the chaos
game, you take a sheet of paper, and draw three points, A, B, and C. Then you pick a point P0, something where inside the triangle formed by A,B, and C. Then you do the following repeatedly, for each iteration i+1:
- Randomly select one of A, B, and C.
- Draw a line segment between the most recent pi and the selected point. Find
the midpoint of that segment. That is point pi+1. Draw a dot on pi+1
Keep doing that, over and over again. If you try this yourself for a few iterations, you’ll see
why it’s called the chaos game: the point jumps around seemingly at random within the triangle. The
set of points that results from this seemingly random process is the attractor for the
dynamical system described by the chaos game. For example, over to the right, I started with P0, and then did iterations with line segments from p0 to A, from p1 to C, from p2 to C, p3 to B, and from p4 back to C.
So what’s the result? What does the attractor for the chaos game look like?
It’s the Sierpinski gasket, distorted so that the triangles are shaped the same as the
triangle defined by A, B, and C:
For any of the L-system fractals, there’s a dynamical system for which the fractal is
Let’s step back just for a moment. What’s going on here? The starting point is the dynamical system.
Just what is a dynamical system?
A dynamic system is basically a kind of differential equation – it’s a rule that specifies a future
state of a system in terms of a previous state of the system. The states in this system are really points
within some kind of topologial manifold. Every dynamical system has something called a state space
– that is, the space which includes the set of points within the manifold that are reachable by some
sequence of iterations of the system.
On some level, the basic study of differential equations – what most of us math geeks studied in our
diffeq courses in college is largely about trying to find closed form equivalents of a given
dynamical system: that is, a solution for the system where you can find the value of the state of a system
at a specified point in time in one step, without needing to iterate through its previous states.
The attractor is the set defined by a dynamical system. What makes it an attractor is the somewhat
odd property that instead of varying
wildly, the attractor will always try to push the system back into its basic space. Its state space
is strongly centered on some basic set of points, and it will always drift back towards those points. Fractal attractors (also called strange attractors) are dynamical systems with a fractal
structure. The chaos game contains embedded within it the self-similarity property of the
Sierpinski gasket; so its attractor set has that structure.
For another fascinating example of this, taken from this excellent explanation of
iterated function systems, here’s a more complex dynamical system. The
way that it works is that there are 4 possible rules for getting the next point, each of which
has an assigned probability. The next rule is selected randomly, according to the specified
|Rule||Probability||Next X||Next Y|
This is the classic Barnsley Fern fractal:
That strange set of equations contains the structure of a fern.
One of the interesting things about the notion of fractal attractors is that one of the objections
to fractals is that they’re too irregular, too strange. Mathematicians tend to like things to be clean and simple. They like continuous, differentiable functions; they like their diffEQs to have closed forms. The idea that most functions are irregular, and that the functions we’ve spent so much time studying are less than a grain of sand on the beach is a depressing one to most people in math. One
of the things that originally caused some resistance to the idea of fractals is that they’re not from the world of clean functions, the world of regularity and continuity that we so adore.
But the fact that attractors include fractals – that out of the universe of iterative systems, nearly
all of them escape – turn into erratic scattering chaos, except for the tiny spec of dust that we call attractors – that tells us that fractals are part of the world of things that we can analyze and understand, even if they seem strange.
We can understand what these iterative functions systems do – how they generate the fractal attractors – by looking at them as affine transformations, and appyling techniques from conventional euclidean geometry to them. Next post, I’ll show you why the chaos game is just another way of defining the