By Gemma Penson - Computer Science Student @ Trinity Hall, Cambridge

At school, students are often taught about predictable phenomena such as chemical reactions, eclipses, and gravity. When modelling scenarios in school, a system of equations, each involving a set of variables, can often be used to gain insight into that scenario. For example, equations involving the variables acceleration, velocity, and distance can be used to predict the time it would take for a ball to hit the ground. If any of the variables were to change, one could predict reasonably how it could affect the outcome.

Predictability is seldom this simple in reality, with chaos theory being the exploration of phenomena that are near to impossible to predict or control such as the stock market, the weather, human brain states and even how milk swirls in your morning cup of tea. Chaos is the science of surprises, of the nonlinear and the unpredictable. It explores the transitions between order and disorder, which often occur in surprising ways.

Chaotic phenomena are facts or situations where there is no discernible regularity or order. They are often described using fractal mathematics, where a factual is an infinitely complex never-ending pattern. They are created by iterating a simple process in an ongoing feedback loop and are self-similar, which means that their properties are preserved with respect to scaling in space or time. One class of fractals is the Mandelbrot set - a set of non-real (complex) numbers that creates some of the most famous images in mathematics.

Fractals are the picture of chaos! They exist between our familiar dimensions but are perceived by everyone on a daily basis. This is because the laws of nature repeat fractals on different scales. For example, trees are natural fractals as each tree branch, from the trunk to the tips, is a copy of the one that came before it.

Chaos is famously illustrated by Lorentz’s famous butterfly effect: the notion that a butterfly stirring the air in Hong Kong today can transform storm systems in New York next month. Chaos implies that systems are very sensitive to changes in their initial conditions - even minute alterations such as a butterfly stirring may cause drastic changes to their outputs. Scientists can never measure the initial conditions of a chaotic system precisely enough to make accurate predictions past a point in time. Minor errors in the initial measurements are sufficient to have dire repercussions on the accuracy of a prediction as the system progresses.

In popular culture, the term “butterfly effect” is almost always misused to describe the idea that small actions can have a significant impact. Many people believe that the butterfly effect can be harnessed in order to achieve one’s goals over time, but the mathematical reality is that miniscule changes in a complex system may have a monumental effect or no effect whatsoever. Knowing which will turn out to be the case is virtually impossible!

__Further reading:__

1. 12th May 2015. Stanley McChrystal. Team of Teams: New Rules of Engagement for a Complex World.

2. 2018. Pearce, Kyle. DIY Genius. (2018). *Fractals In Nature: Develop Your Pattern Recognition Skills In The Forest*. [online]

3. 2018. Fractalfoundation.org. *What is Chaos Theory? – Fractal Foundation*. [online]

*4. *1st August 2019.* Seeker. How Chaos Theory Unravels the Mysteries of Nature YouTube*. [online] Available at: https://www.youtube.com/watch?v=r_5shyQGIeA