# School of Mathematics

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“Phil Routledge Allerton GrangeThe Cryptic Challenge STEM session introduced areas of great interest and enjoyment and gave students examples of careers involving maths.

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Within the School of Mathematics we are currently holding a number of practical demonstration kits, developed by the Institute of Mathematics and its Application (IMA) which are available to borrow. We have 5 kits available each helping to promote an area of mathematics in a hands-on, practical way which would be great for in-class demonstrations and for use at fairs.

### Galton Board: A Practical Demonstration of Probability

Most people have some awareness of chance and probability. The probability of an event occurring can be assigned a number between 0 and 1; if the probability of occurrence is 1, then the event is certain to happen. If it is zero, it will never occur. Naturally, most events will have a probability somewhere in between. Since a die has six faces, if it is fair, the probability of throwing any particular number is 1/6. If you toss a coin, the probability of a head is ½. What happens if you toss a coin ten times? Will you get five heads and five tails? Suppose you get ten heads; what is the probability of getting a head on the next throw? ... read more

### Soap Bubble Wires: The Energy of Bubble Films

The soap bubbles we normally see are beautiful and of all sizes, but only one shape: they are all spherical. The reason is that this is the shape that contains the pressurised air within, with the least surface area, a fact which can be shown mathematically. This is the result of the molecules within the soap film acting to find the shape which will minimise the energy stored in the film.

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### Double Pendulum: The mathematics of chaos

The mathematics of chaos is becoming more familiar and many people will have heard of the ‘Butterfly Effect’, used as an illustration of the sensitivity of subsequent weather conditions to very small changes of initial conditions. Generally speaking, in a moving or dynamical system, if a non-linearity is present (that is, a graph of cause and effect would not be a straight line) and the system is sensitive to initial conditions, then its state after a period of time will be impossible to predict. But what does this mean in practice?....Read more

### Aerodynamics: Aerofoil Demonstration

An aerofoil is essentially an aeroplane wing section, but it can also be found in performance cars to provide down-thrust, and therefore help the tyres grip better. Other applications include fans, propellers and even wind turbines. The aerofoil demonstration comprises a variable speed, propeller generated, collimated airflow which passes over an aerofoil.... Read more

### The Travelling Salesman

It’s a common problem, and applies as much to travelling salesmen as to paper rounds! The salesman has a widely spread number of calls to make, but wishes to do so by covering as little ground as possible. Should he travel in a closed shape like an irregular polygon, or would it be better to criss cross the area to shorten his journey? In times of high fuel costs and concerns over carbon emissions, this is more than just an intellectual puzzle!... Read more

### Harmonograph

You might think that a pendulum is a simple thing, swinging back and forth in a predictable way, and if it had a pen attached to it, then it would produce nothing more than a straight line. If however you could attach a second pendulum to your pen, swinging at right angles to the first pendulum then you start to produce beautiful and surprisingly complex drawings. Although patterns appear initially to repeat, as the energy stored in the swinging pendulums is expended in the bearings of the machine, the size of the pattern gradually decays, increasing the beauty and complexity of the figures and providing an extra dimension for learning.....Read more

### The Penrose Tiles

These two shapes have a remarkable property. They can tile a plane but not periodically. In other words the tiling of the plane they produce is not made of a single patch of tiles repeated.

For further information about these kits and how to borrow them please contact:

Jessica Brennan

Tel: 0113 343 5116

Email: j.m.brennan@leeds.ac.uk

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