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The Chaos Treasure Hunt

by Professor Chris Budd & Professor Giles Hunt

The Chaos Treasure Hunt shows how maths can be combined with an understanding of the way that things bend, to learn about the shapes found in rock cliff faces. Sometimes maths predicts that naturally occurring things have very regular patterns, but paradoxically it can also predict that nature can behave in an almost random, chaotic way. We have to face this paradox every time that we use maths to describe the natural world, and this is the essence of this film. It looks at two examples of using maths in this way, to study rocks and also to study a chaos pendulum. Some of the chapters (in particular Chapters 10, 11 and 13) go quite deeply into the mathematics and use ideas (from calculus) that would be difficult for many young people at school to understand. However, the film has been put together in such a way that these chapters can be omitted without losing the main thread of the story; they can be viewed later if you want to learn more detail about the underlying mechanics.

The following gives specific notes for each of these chapters.

Chapter 10. In this chapter we describe the equations that describe a bending beam. These equations are a mathematical short-hand for the process of balancing the forces applied to a single beam, with those generated internally when bending it. The equations turn out to be fourth-order differential equations, which are normally beyond the scope of what can be done at school level, so we wouldn’t really expect young people to understand how they are solved. However, the film does give an idea of what is involved in beams under bending forces, and some of the parameters involved. E for example is described as material stiffness, also sometimes referred to as Young’s modulus. The symbol I is linked in the film to the thickness of the beam. It is usually referred to as the second moment of area and depends on a number of characteristics of the cross-section, including the thickness. In fact it varies as the thickness cubed of the beam, so doubling the thickness for example would increase the resistance to bending by a factor of 8. The symbol q is used to represent the load, and the equation is described in terms of a fourth derivative of the displacement w. To find the actual shape of w, this equation would need to be integrated four times.

Chapter 11. Shows a discussion between a mathematician and an engineer. In some ways the most revealing part of this chapter is not the details of the science, but the way it shows two scientists interacting, and also shows that science is never cut and dried, but often involves discussion and argument. The discussion looks at a link between the rock folding problem and the way that light (and water) waves develop. This illustrates a key idea in science, that ideas from one field can often be used in another, and that maths is the common language that makes this all possible. The principle in question is due to Huyghens and was derived in the 17th century to look at the motion of a wave front. It is used all the time in the study of optics, but as far as we know has never been applied to geology before. It’s an interesting and perhaps not completely unrelated fact that Huyghens was also responsible for inventing the pendulum clock.

Chapter 13. The double pendulum is an excellent demonstration of the concept of chaos, that is really good for showing to young people. It comprises two simple pendulums coupled together, jointed very much like your leg with one pendulum being the lower half of the leg, the other the upper half and your knee acting as the joint between them. The pendulum shows that a simple system can have very complicated behaviour, and that its behaviour depends crucially on how you start it off. In the film we show that the pendulum can have very predictable, periodic motion (with the two parts either in phase or out of phase), or completely
unpredictable chaotic motion. When it behaves chaotically it is impossible to tell exactly what it is going to do next. Remarkably, whilst the individual components of the pendulum are very simple, the way that they are coupled together makes the overall motion far more complex. In this way the pendulum closely resembles the behaviour of much more complicated systems such as the weather.

In the film we show the actual mathematical equations which describe the motion of the pendulum. These are much too hard to solve by hand, but a computer can solve them and show how the pendulum behaves. In this way we show an important principle, that by combining equations and a computer we can simulate really complicated things. This idea works just as well when we look at the much harder problem of seeing what patterns we get in rocks.

Chris Budd & Giles Hunt
March 2007

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Funded by EPSRC © University of Bath 2007