Total, Utter CHAOS

By Isobel Abell, Class of 2020.

Bourke Street on Christmas Eve. Eight dogs in a small room after you say “Walkies!” The state of online learning. We’d all describe these situations as chaotic, but what is chaos? Collins Dictionary describes chaos as a “state of complete disorder and confusion.” While this perfectly describes my undergraduate degree, it’s not quite enough for mathematicians.  Time dependent systems are classified as chaotic when, once we change the initial conditions, the outcome changes drastically and unpredictably.

Edward Lorenz, the father of chaos theory, elegantly summarises this idea:

Chaos: When the present determines the future, but the approximate present does not approximately determine the future

On the surface chaos may seem completely random and indecipherable but looking closer we find underlying patterns. This is what keeps chaos theorists up at night: the quest to search for order in disorder.

Double the pendulum, double the fun

The simplest way to picture chaos is by looking at pendulums.

Stare at the grandfather clock and the end of the hall and you’ll notice its arm swings monotonously back and forth as you fall into a deep, deep sleep… The unique balance of momentum and gravity mean the pendulum swings side to side and would continue to do so forever if not for pesky things such as friction and air resistance. No chaos here.

However, add another weighted arm to the bottom of your pendulum and you’ll get a double pendulum. Congratulations, you’ve created chaos. Dropping the pendulum from different heights, changing the length of an arm or changing the weights transforms what happens. We no longer have a simple oscillating system but an unpredictable and intricate arrangement that sketches out beautiful arcs like the ones below.

If you’d like to play with a double pendulum yourself, check out this simulation.

Let’s zoom in on fractals

Now we’ve considered the double pendulum, let’s get a little more abstract with fractals.

The simplest fractals are generated by repeating a simple process and graphing the results at each step. Certain processes can be deemed chaotic as they produce vastly different patterns when starting with a different arrangement. These are the fractals: never-ending patterns. The hunt for order in these intricate configurations leads to extraordinary discoveries. For example, in the fractal below zoom in enough times and you’ll return to the pattern you started with. What seems random and inexplicable has suddenly becoming infinitely repetitive.

So why do we care about something as abstract as fractals? Benoît Mandelbrot (after whom the above fractal is named) says in his book The Fractal Geometry of Nature:

Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line.

Fractals appear everywhere in nature, from the webbing on leaves to the paths rivers trace on their journey to the sea. On first glance these things may seem unregulated and indecipherable, but the genius of nature only becomes clear once we allow ourselves to zoom in.

Chaos theory allows us to make sense of the ever-changing world around us. We can dive head-first into complexity itself and discover an intricate world of organised patterns completely invisible to those on the surface. To me, this is the beauty of mathematics: an infinite world begging us to explore its secrets.