## 3D Geometry & Regular Polyhedra

*An interactive activity for Year 9–11 students by Dr Robert Whittaker.*

There are five regular polyhedra, known as the Platonic Solids. In this activity, students will be given plastic shapes to build their own polyhedra, and to discover the five Platonic Solids for themselves. With guidence from the instructor, they will understand why these five are the only possible regular polyhedra, and investigate the properties that they have.

Older students can also go on to consider the semi-regular polyhedra, which have faces of more than one type of regular polygon, but the same sequence of different faces around each vertex. These comprise two infinite sets of prisms and anti-prisms, together with the 13 Archimedean Solids.

## “Who’s the best?” Mathematics in Sports Ratings

*A talk by for year 10-13 students by Dr Jonathan Kirby.*

What are the odds of Norwich City winning the league this year? Will Andy Murray be the world number 1 tennis player? Will he or Jo Konta win Wimbledon? Mathematics is used to measure success in sport and also to predict future performance. What do these numbers mean and where do they come from? We will explore how odds and probabilities can be used to make predictions and to create world rankings.

## Goodstein Sequences

*A talk for students in years 11–13 by Dr. David Aspero.*

Description: Given natural numbers \(n\) and \(m\), the hereditary base-\(n\) representation of \(m\) is obtained by writing \(m\) in base \(n\), writing each exponent in this representation again in base \(n\), and so on. For example,

\[18=2^4+2^1=2^{2^2}+2^1=2^{2^{2^1}}+2^1 ,\]

which shows that the hereditary base-2 representation of 18 is \(2^{2^{2^1}}+2^1\). The Goodstein sequence starting at \(m\) is obtained as follows: Its first member, \(G(m)(1)\), is \(m\) itself. The second member, \(G(m)(2)\), is obtained by replacing each 2 by 3 in the hereditary base-2 representation of \(m\) and subtracting 1. The third member, \(G(m)(3)\), is obtained by replacing each 3 by 4 in the hereditary base-3 representation of \(G(m)(2)\) and subtracting 1. And so on.

For example, the Goodstein sequence whose first member is 4 starts \[4, 26, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, 401, 458, \ldots .\] We will generate some Goodstein sequences and will see that they can be very long and and grow extremely fast. Surprisingly, though, every Goodstein sequence eventually reaches 0.

I will talk about ordinals, which are infinite counting numbers stretching beyond the natural numbers, and will sketch some ideas on how to use them in order to prove this surprising theorem. I will also mention some striking connections of this theorem with other parts of mathematics.

## Maths in a Box: Pythagorean Triples and Beyond

*A talk for years 11–13 inclusive by Dr Mark Cooker.*

If a right-angled triangle has sides of lengths *a*, *b*, *c* where *c* is the hypotenuse, then Pythagoras's Theorem says that *a*-squared plus *b*-squared equals *c*-squared. If *a*, *b*, and *c* are all whole numbers then (*a*,*b*,*c*) is a Pythagorean Triple. A famous and ancient example is the (3,4,5) triangle. Another is (5,12,13) and there are more. In this talk we will show how to make a different Pythagorean Triple for every person in the audience! Some of these ideas carry over to thinking about a rectangular box. We will look at the problem of finding boxes which have edges and face-diagonals all of which are integer lengths. We end the talk with a 250-year old problem which still has not been solved.

## How not to Lose your Mind using Group Theory

*An interactive activity for years 11-13 inclusive, by Dr Robert Gray.*

The Futurama Theorem is a result in an area of algebra called group theory, that was devised for the episode 'The Prisoner of Benda' of the TV show Futurama. The theorem was created and proved by show writer Ken Keeler, who has a PhD in mathematics. In the episode Professor Farnsworth builds a machine so that he and Amy can switch minds with each other. As the episode progresses, other characters from the series find the machine, and they all engage in a mind swapping frenzy. However, when they decide that they all want to go back to having their minds in their own bodies, they discover that the machine cannot be used twice on the same pair of bodies. The Futurama Theorem concerns the question of whether, even with this restriction on the way the mind swapping machine operates, it is possible to find a sequence of swaps that will restore everyone's minds to their correct bodies.

In this activity, we will build our own mind swapping machines and use them to investigate this problem, revealing how it relates to ideas from group theory.

## Who's got your Number?

*A talk or interactive activity for years 11-13 inclusive, by Prof. Shaun Stevens.*

What keeps credit card numbers safe when they are transmitted across the internet? The answer is that they are encrypted using Mathematics, with the key to many types of encryption being properties of prime numbers and the inherent difficulty of factorizing large numbers. In this talk, we will look at one Public Key encryption method, called RSA: how it works (including encrypting our own message using a simple version of it) and why, when used sensibly, it is safe? for now, at least.

## Can Irrationals Repeat?

*A talk or interactive activity for years 11-13 inclusive, by Prof. Shaun Stevens.*

When fractions are written as decimals, they always end up with a repeating pattern: for example, 1/4=0.250000… and 23/300=0.076666…. The converse is also true: any decimal which eventually repeats can be expressed as a fraction. Thus *irrational *numbers (numbers which are not fractions) like √2=1.414213… have decimal expansions with no repeating pattern.

However, there is another way of writing real numbers, connected to cutting up rectangles into squares as in the picture. We will discuss this (and maybe cut up some paper), examining which numbers end with a repeating pattern, as well as some related problems which are still unsolved.

## Levers, Archimedes' Principle and Infinitesimals

*A talk for years 11-13 inclusive by Dr Jonathan Kirby.*

Whenever we open a door we are using a lever to move a heavy object with a small effort. Archimedes recognised this principle and declared that if you gave him a place to stand and a long lever he could move the whole world. But is that true? Might the world be infinitely heavy? And if it were, would that mean Archimedes was wrong?

We will explore levers and ratios of numbers, and see how Archimedes principle means there can be no infinitely large or infinitely small numbers. On the other hand we will see that infinitely small numbers can make sense too, and we will learn how to add, subtract, multiply and divide with them.

## Fundamentals of Resonance

*A talk for year 12 or 13 students interested in physics and applied mathematics by Dr Davide Proment.*

Resonance is an incredibly simple but fundamental mechanism arising in Nature. It can be easily introduced to students using tuning forks and resonant chambers, but its applications are endless.

In this talk I will present the fundamental model of resonance consisting of a source, wave signal, and resonator as a receiver. I will then discuss some everyday applications and consequences of the resonance mechanism. Examples will include radio signals, energy transfer from the Sun to the Earth, microwave cooking, and oscillation of buildings. The talk will end with some comments on how physicists routinely use the resonant mechanism to detect fundamental particles like the recently-discovered Higgs boson.

## Sequences and Series

*A talk or interactive activity for year 12 or 13 students, by Dr Vanessa Miemietz.*

If you have a sequence of numbers, what happens when you try to add them all up? Does this make sense, or, under what conditions does this make sense?

The answer to this lies in questions of convergence. We will look at what this means and discuss some (possibly surprising) results.

## Forecasting the Weather: Why is it so Difficult?

*A talk for year 12 or 13 students by Prof David Stevens.*

It is possible to write down equations that govern the dynamics of the atmosphere. These equations are complex and difficult to analyse in detail. Weather forecasting involves numerical solution of the equations. In principle weather forecasts could be generated for many days, weeks or even months ahead. However forecasts are only skilful for a few days. This is because the atmosphere is a chaotic system. It is possible to demonstrate the difficulty using an analogue featuring equations of the kind studied in A-level Further Mathematics (FP2 Edexcel, FP3 OCR and AQA). Improving forecasts and identifying when forecasts break down is an exciting area of research involving mathematicians and meteorologists.

## Fluid Motions and Turbulence in Real Life

*A talk for year 12 or 13 students **by Dr Davide Proment.*

Fluids are almost ubiquitous in our normal life: the air that we breath, the water that forms approximately 65% of an adult human body, the coffee or tea that we drink every morning. The study of fluid motion is a challenging topic for mathematicians: despite the fact that most of the interesting fluid phenomena could be reproduced and observed by the naked eye in a lab, their full mathematical understanding and forecast is still an open problem.

In this talk we will present the main mathematical model of fluid motion — the Navier–Stokes equation — and explain the historical contributions of Lenoardo da Vinci, Claude-Louis Navier, Sir George Stokes, and Osborne Reynolds. We will then show how phenomena like the von Karman vortex sheet, the Kelvin-Helmoltz instability, Rayleigh-Bénard convection and turbulence are easily seen in real life, and discuss their technological applications.

## Sonic Booms and Solitons — or why hyperbolic functions are important

*A talk Year 12 or 13 students by Dr Paul Hammerton.*

When an aircraft is going faster than the speed of sound, a sonic boom is produced. This is sometimes heard on the ground and sounds like a double crack of a whip. What we hear is a sudden increase in air pressure over a very short time. If a boat comes to a sudden stop in a canal or narrow river a single 'hump' of water can be formed which travels unchanged over long distances. This is known as a solitary wave, or soliton. Waves of this form occur in lots of other fields, including information transfer in optic fibres and in neuroscience.

What do these two types of wave have in common? Everyone studying A-level mathematics knows about trigonometric functions and what they can be used for. Closely related are the hyperbolic functions, and these underpin the mathematics behind sonic booms and solitons. In this lecture we start from scratch with the definitions of the hyperbolic sine and cosine functions, establish some of their properties, and then focus on how they can be help describe several physical phenomena.

*Some familiarity with the exponential function before the talk would be an advantage, but no prior knowledge of hyperbolic functions will be assumed.*

## Aerodynamics of Sport

*A talk for Year 12 or 13 students, by Dr Paul Hammerton.*

Aerodynamics plays an important role in many sports — from the high technology of cycle helmets and F1 car design to how a footballer bends a free-kick around a defensive wall. In this lecture we will focus on the forces acting on a body moving through the air. The theory is similar for a high speed car or a ball. We will look at the effect of changing the shape of a car, but mostly we will be looking at how balls move through the air. What is the reason for dimples on a golf ball? Why does a cricket ball sometime swing through the air?

No knowledge of Physics or Mechanics is assumed.

## Different Sizes of Infinity

*A talk for year 12 or 13 students, by Prof Mirna Dzamonja.*

Infinity is a magical concept which enters human imagination through philosophy, art but also mathematics where it plays an important role of modeling natural processes that happen over large time or in a very large domain. In fact, understanding infinity in mathematics is necessary in both abstract and applied contexts, starting from the fact that many engineering applications of mathematics use infinite series to approximate calculations. But what exactly is infinity? Is there only one? How can we measure it? Can we do mathematics with infinity? This and other questions can be addressed in a very beautiful way, invented in the 19th century by George Cantor, and continuing to interest mathematicians and others up to this day.

## Chaos, and How to Create It

*A talk for year 12 or 13 students, by Dr Mark Blyth.*

In Greek mythology Chaos meant the void that existed at the beginning of time. In Mathematics, chaos refers to the apparent randomness and disorder which can appear, seemingly out of nowhere, in calculations ranging from the very simple to the very complex. Chaos in weather prediction popularised the idea of the 'butterfly effect', which states that a butterfly flapping its wings in Moscow can lead to a thunderstorm in New York. In this talk we will create chaos on a pocket calculator by following a strikingly simple set of rules. In our calculations we will discover how, hidden within the chaos, we can find astonishing structures and patterns.