I was very excited earlier this week to receive a suggestion from one of my good friends about what to write for this column. What’s even cooler is that their suggestion lined up pretty well with a topic I wanted to write about but didn’t have a good example for—thanks to this friend, now I do!
The topic is chaos theory, which I’ll describe in a little bit, and the example is pendulum painting, which has recently become a popular art form on TikTok. Pendulum painting has a pretty self-explanatory name. You start with a blank canvas and hang a pendulum over it. This pendulum is a container of paint with a small hole; you let it swing, and the paint gradually falls onto the canvas as the string loops back and forth. The final product has a cool, almost Pollock-like quality to it.
These paintings are beautiful, but what fascinated me is that a pendulum is used. Many of you have likely studied or will study pendulums in a class. They are approximately harmonic oscillators, meaning that the net force applied to them (in this case, gravity) is directly proportional to the displacement. This approximation is not always valid, but all pendulums have a periodic swing to them like harmonic oscillators do.
However, our system involves a change in mass as the paint falls out of the container. It would be very difficult, even if we had information about all the paint particles and the position and speed of the container at every point in time, to predict exactly what the final painting would look like on the first try. In other words, even a slight change in the initial conditions—the type of paint used, the starting position of the pendulum, and the environment where you’re painting—could have a huge effect on the resulting painting.
Mathematicians refer to these types of systems as “chaotic,” and seek to study them through the lens of “chaos theory.” The goal of this theory is to recognize patterns that will arise in such systems. The dynamics may look vastly different at first when the initial conditions are slightly altered but eventually can fall into a common equilibrium. The mathematical name for this equilibrium is an “attractor,” because it attracts many different starting conditions to the same end result.
Chaos theory plays a major role in modeling systems in the real world. Another chaotic system is the dynamics of three or more objects in space—our solar system, for instance. We are lucky that this system has found a stable attractor. It’s also been shown that chaos can arise in population dynamics. Luckily for humans, we have had relatively stable population growth throughout our history.
Less stable systems that exhibit chaos are, for example, Earth’s climate and financial markets. In the first case, meteorology relied on mathematical models to try and best predict the weather. Models fail when information is left out, which is why data collection is so important for meteorologists.
In the second case, chaos arises when an economic bubble occurs. It is difficult to identify bubbles when they first appear, but perhaps chaos theory could help do that in the future, although there is debate on whether this is the case. Economists also use analysis based on this theory to study business cycles and shocks to the market, such as the COVID-19 pandemic.
In all these cases, there’s always a gap between what we can know about the dynamics and what actually occurs. Despite this, chaos theory assures us that patterns are still possible, and very often do occur. So, we can still have beautiful pendulum paintings! But we aren’t able to predict everything. In this case, or in many others, it’s artistic prowess and human ingenuity that melds chaos and beauty.
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