Essence of Chaos: Experiments and simulations with balls and tubes

2020/09/10

Hi! My name is Abram Hindle and I’m a TWOSE Science Fellow. One of my favourite science books is Edward Norton Lorenz’s “The Essence of Chaos” Lorenz was a mathematician and meteorologist who was frustrated by modelling weather on computers. He found that the weather was a dynamic system full of patterns that we can expect and name—such as thunderstorms—but some of this behaviour was hard to predict very far into the future. Currently we’re quite good at estimating if it will rain or snow today but our further predictions of rain or snow a week from now are much poorer. Lorenz wanted to understand why.

In studying why, he founded chaos theory. Chaos theory studies why some systems would act unpredictably under certain conditions and very predictably under other conditions. By predictable Lorenz means that you estimate accurately the result. Often chaotic systems exhibit repeated patterns but how and when they do this is often unpredictable. Turbulent water, the mixing of dirt into water, the patterns of oil on water are examples of chaotic systems whereby we know the patterns we’ll see, but what actual pattern might be really difficult to predict. A big takeaway from chaos theory is that systems composed with simple rules, depending on initial conditions, could be unpredictable.

His book, “The Essence of Chaos” is full of these examples and simulations of these behaviours. As a software developer / computer scientist myself I am drawn to the simulations, as they demonstrate simplified versions of nature exhibiting the complexities of patterns in nature. As a programmer it is very satisfying make programs that mimic nature without a lot of effort. One chaotic experiment in the book is the simulation of a skier sliding down a bumpy hill full of moguls. We can replicate this experiment to demonstrate predictability given initial starting conditions.

Experiment at home

How can you play around with chaos theory at home? Well we can run an experiment with tubes and balls. We can see how changing the steepness of the tube can create behaviours that are hard to predict. Our main behaviour will be the interaction of a ball rolling down a tube, and we’ll try to guess where it will exit. We’ll find that different steepness are easy to predict while other steepnesses result in more variation in the exit points of the ball.

A ball rolling down a surface or tube.

A game with prediction you can play requires:

We’re going to drop the ball or marble down the tube and guess where the ball will exit the tube.

So the game to play is to record the angle that your surface is at, and the distance from the final starting position to the exit position (where it flies out off of the tube or surface). Do this 5 to 10 times for a each steepness (lots, a bit, nearly flat). Reliable experiments in probability typically repeat the trials many times.

Look at the minimum and maximum distance from the center of the tube where the ball rolls out and answer these questions:

If you can answer the first question you know the conditions for creating a chaotic system.

If you can answer the second question you know the conditions for that make for a predictable system.

But in both cases you are using the same materials and the same system. This is Lorenz’s point. Some initial starting conditions (angle of surface)

Simulate at home

What drew me to Lorenz’s book is that I like to program. I like to tell the computer what to do and I like to simulate experiments on the computer. Personally I am also very lazy and I would prefer if I could provide instructions to the computer to get it to repeat hard work for me, such as the experiment I laid out above.

I coded up a simulation of this experiment, it can simulate 10s, 100s, and 1000s of balls rolling down a half tube (half pipe) all on a web page. You have control over where the ball starts on the top edge of the tube.

Book information

https://uwapress.uw.edu/book/9780295975146/the-essence-of-chaos/

Title: The Essence of Chaos

By Edward N. Lorenz

    PUBLISHED: October 1995
    SUBJECT LISTING: Science and Technology Studies
    BIBLIOGRAPHIC INFORMATION: 240 Pages
    ISBN: 9780295975146