Order in Randomness: Fractals and Chaotic Systems


By Steven Gimbel, Ph.D., Gettysburg College

Systems like the weather are sensitive to small changes in initial conditions and are unstable. These are incredibly complex systems. But chaos theory can reveal the order in the randomness.

Board with multiple mathematical formulas and diagrams.
All kinds of systems can be represented using formulas and equations. The ones for chaos theory are much more complex. (Image: Marina Sun/Shutterstock)

Looking for Order in Chaos

Physicists like to order. Their job is to look at intricate systems and find ways to represent them as simply as possible with equations. Small deviations from the large-scale order are dismissed as noise. Only the large-scale regularities are given attention. There is even a joke about a farmer who hires a physicist to design a chicken coop: the physicist begins by assuming spherical chickens.

Chaos theory demands that scientists think differently, that they pay attention to the actual shape of the chicken. That is, they should heed these micro-deviations that they prefer to ignore. In fact, when physicists began to focus on them, something fascinating revealed itself.

When we have a periodic system, the prediction is easy. If Halley’s Comet comes around every 76 years and it was last seen 75 years ago, get ready. The behavior of chaotic systems is non-repeating. There are no steady-state solutions, no periodicity, and seemingly no predictability.

Learn more about the standard model of particle physics and the general theory of relativity.

Non-Repeating Large-scale Order

But it’s not random. When scientists began charting the development of chaotic systems, large-scale regularities did begin to appear in the results. The evolution of these physical systems over time would not return to the same state, but when the states were mapped on a mathematical structure called phase space, patterns began to appear.

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If the chaotic systems were truly random, then the system’s path through phase space would be like that of a drunk wandering down a street. You never know where he may weave and stagger next. There’s no place to stand without risk of being bumped into by the inebriated fellow. In a truly random chaotic system, there is no state which the system might not occupy, and any state could lead to any other state.

Strange Attractors and Fractals

When we plot out the development of the chaotic system, there isn’t a simple map we see in predictable, stable systems. Yet there is also not random craziness either. The line that is the representation of the system’s time evolution could be complicated in odd ways, but it will take definite forms.

It might be a torus, a donut shape. It might look like the wings of a butterfly. It might resemble a saddle. The path of the system represented in phase space will stay on these surfaces and eventually fill them in. Mathematicians call these surfaces ‘strange attractors’.

3D model of the Lorenz attractor, a famous 'strange attractor' that models a chaotic system.
The uncanny butterfly shape of the Lorenz attractor, one of the most famous ‘strange attractors’. (Image: local_doctor/Shutterstock)

The mathematical structures of their surfaces are the most popular element in chaos theory: fractals.

A fractal is a shape that is self-similar when you look at different scales. If you look at a fractal pattern and zoom in on any segment, that segment will look like the larger section you started with. We can think of one of the simplest fractal patterns, the Koch curve also called the Koch snowflake, which named for its originator, the Swedish mathematician, Helge von Koch.

A Simple Fractal: the Koch Snowflake

Start with an equilateral triangle, that is, a triangle with three equal sides. Now take each side and use the middle third as a side for another equilateral triangle. This means that each of the three sides of the triangle now has a triangle sticking out of it. The shape is the familiar Star of David.

But now take the two outer edges of each of those six triangles and use the middle third of them as the edge of yet another equilateral triangle. Now it looks like a snowflake. You can take all of these little triangles and again build more triangles off of them again, and again, and again, infinitely many times. The result is the Koch snowflake.

Image showing the construction of the Koch Snowflake, a simple fractal.
The first four steps in creating a Koch snowflake. By the fourth step, you can see the snowflake shape. There is an infinite number of steps possible. (Image: Alejo Miranda/Shutterstock)

It is a strange shape. When we went from the original triangle to the Star of David, we increased the perimeter of the shape. It’s a longer path around the Star of David than around the original equilateral triangle. And for each iteration, each addition of new triangles, the length of the perimeter of the snowflake grew a finite amount.

Learn more about the shocking discoveries of non-Euclidean geometries.

Infinite Perimeter, Finite Area

Since we did this an infinite number of times, each increasing the length of the perimeter a finite amount, the result is that the length of the perimeter is infinite. However––and this is one of the weird things––the area inside is finite. We have an infinite line closing on itself and enclosing a finite region. If I had a circle of infinite circumference, I would have an infinite area inside, but the Koch snowflake has an infinite perimeter with a finite interior area.

But the interesting aspect for us is that because the way it’s constructed, any place you look at the Koch snowflake, no matter how close or how far you are from it, it still looks the same. This self-similarity across different scales is the hallmark of fractals. We find this sort of geometry in all sorts of irregular, but not random places in nature. Leaves on trees, cracks in the earth, coastlines, the internal structure of crystals, all show fractal geometry.

Fractals to Model Nature

Since we find fractals in nature, we also use them to model nature, not just scientific models like those we use to predict the weather, but also in art. Since 1982, artists have been using fractals to create more realistic landscapes in the computer-generated graphics for the backgrounds of movies.

Abstract computer-generated fractal shape. Chaotic systems have underlying patterns.
A computer-generated fractal mimics the natural pattern of a lettuce leaf. Many fractal shapes appear strikingly similar to the shapes and patterns found in nature. (Image: DasArts/Shutterstock)

The masters at Pixar developed the technique for Star Trek II: The Wrath of Khan. In the animated sequence that introduces the central plot device of the film––the Genesis Effect, which makes uninhabitable planets into lush areas suitable for life––the transforming planets are made real by the first use of fractal drawing in film. It’s since become a standard tool in the graphical design toolbox.

Learn more about the underlying reality that governs the universe.

Order in Complexity

Reality seems more real to us when it is modeled on chaos. Science seeks order in nature and finds it, but that order is so complex that it seems to give rise to disorder. But in examining the disorder of chaotic systems closely, we find a new order.

Chaos is not randomness, but a complexity within the universe we never thought was there. Our attempts to make the world comprehensible at first led us to demand that it be regular, simple, periodic, predictable. Nature said no and we had to change our image of it. But it does not mean that science is useless; on the contrary, it’s the science that gave us this richer knowledge.

Common Questions about Fractals and Chaotic Systems

Q. What is a ‘strange attractor’?

‘Strange attractor‘ is a term given by mathematicians to a surface created in phase space by the development path of a chaotic system. The path will stay on these surfaces and eventually fill them in.

Q. What is a fractal?

A fractal is a shape that remains self-similar at all scales. This means that if you look at a fractal pattern and zoom in on any segment, that segment will look like the larger section you started with.

Q. Are fractals natural or artificial?

Fractals can be found everywhere in nature, from leaves on trees to cracks in the ground. But fractal shapes can also be artificially created using computers.

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