Index of Feature Articles

The computer animation used in movies and games is now so lifelike, it is very hard to believe that you are actually watching a surface built from simple shapes of triangles. Phil Dench tells Plus how he uses mathematics to help bring these models to life.

As an electronic musician Oli Freke has always been fascinated by sine waves, so much so that he's created a song based on them for the Geekpop festival, which is currently taking place on the Web. In this article he explores his song, touching on ancient Greek mythology, strange piano tunings and Johann Sebastian Bach.

In 1979 decorating work in a house in Vienna revealed a set of medieval frescoes depicting a cycle of songs by a 13th century poet, who was particularly fond of satirising the erotic relationships between knights and peasant maidens. The frescoes are of great historical significance, but they are badly damaged. In this article Carola Schönlieb explores how mathematicians use the heat equation to fill in the gaps.

Describing the motion of fluids is a huge and unsolved mathematical problem. There are equations that seem to describe it well, but their complete solution is way beyond reach. But could there be a simpler method? The physicist Jerry Gollub tells Plus about a new discovery which combines experiment with sophisticated maths.

The prime numbers are the atoms amongst the integers, and while we know that there are infinitely many of them, there's no general formula that generates them all. Julian Havil looks at a little-known algorithm that sieves out all primes up to a given number, and which is astonishing in its simplicity.

It's International Year of Astronomy and all eyes are on Galileo Galilei, whose astronomical observations 400 years ago revolutionised our understanding of the Universe. But few people know that Galileo wasn't the first to build a telescope and turn it on the stars. That honour falls to a little-known mathematician called Thomas Harriot, who excelled in many other ways too. Anna Faherty takes us on a tour of his work.

With the credit crunch dominating the news, columnists have been wailing about "chaos in the markets", and "turbulent" share prices. But what does move the markets? Are they deterministic, or a result of chance? Colva Roney-Dougal explores the maths, from chaos to group theory.

Mathematics takes to the stage with A disappearing number, a work by Complicite, inspired by the mathematical collaboration of Hardy and Ramanujan. Rachel Thomas went to see the play, and explains some of the maths. You can also read her interview with Victoria Gould about how the show was created.

In the light of recent events, it may appear that attempting to model the behaviour of financial markets is an impossible task. However, there are mathematical models of financial processes that, when applied correctly, have proved remarkably effective. Angus Brown looks at one of these, a simple model for option pricing, and explains how it takes us on the road to the famous Black-Scholes equation of financial mathematics, which won its discoverers the 1997 Nobel Prize in Economics.

If you like mathematics because things are either true or false, then you'll be worried to hear that in some quarters this basic concept is hotly disputed. In this article Phil Wilson looks at constructivist mathematics, which holds that some things are neither true, nor false, nor anything in between.

When it comes to describing natural phenomena, mathematics is amazingly — even unreasonably — effective. In this article Mario Livio looks at an example of strings and knots, taking us from the mysteries of physical matter to the most esoteric outpost of pure mathematics, and back again.

We live in a world full of information and it's a statistician's job to make sense of it. In this article Dianne Cook explores ways of analysing data and shows how they can be applied to anything from investigating diners' tipping behaviour to understanding climate change and genetics.

What's your strategy for love? Hold out for The One, or try and avoid the bad ones? How long should you wait before cutting your losses and settling down with whoever comes along next? John Billingham investigates and saves the national grid in the process.

According to one mathematician, god created the whole numbers, with everything else being the work of humanity. Why, then did god not equip us with a good way of writing them down? Chris Hollings reveals that our number system, much used but rarely praised, is in fact a work of genius and took millennia to evolve.

Peter Markowich is a mathematician who likes to take pictures. At first his two interests seemed completely separate to him, but then he realised that behind every picture there is a mathematical story to tell. Plus went to see him to find out more, and ended up with a pictorial introduction to partial differential equations.

Much criticism has been levelled at the US voting system, and with this being election year, we're bound to hear more of it. In this article Steven J. Brams proposes an alternative voting system that could help make things more democratic.

Computer-generated art is on the rise, and with it comes a further blurring of the boundaries between maths and art. Lewis Dartnell looks at some stunning examples.

Not so long ago, if you had a medical complaint, doctors had to open you up to see what it was. These days they have a range of sophisticated imaging techniques at their disposal, saving you the risk and pain of an operation. Chris Budd and Cathryn Mitchell look at the maths that isn't only responsible for these medical techniques, but also for much of the digital revolution.

What's the nature of infinity? Are all infinities the same? And what happens if you've got infinitely many infinities? In this article Richard Elwes explores how these questions brought triumph to one man and ruin to another, ventures to the limits of mathematics and finds that, with infinity, you're spoilt for choice.

Richard Elwes continues his investigation into Cantor and Cohen's work. He investigates the continuum hypothesis, the question that caused Cantor so much grief.

In the movies mathematicians are mostly mad. Since here at Plus we firmly believe in our sanity, we're puzzled as to why. So we charged Charlotte Mulcare with the unenviable task of sifting through five well-known maths movies and speculate towards an answer.

The primes are the building blocks of our number system, but there's no general formula that will give you all of them. If you want them, you have to hunt them down one by one. Abigail Kirk investigates a method that does just that.

Neil Pieprzak tells the fascinating story of Andrew Wiles who, with intense devotion and in secret, proved a deceptively simple-looking conjecture that had defeated mathematicians for almost 400 years.

Alan Turing is the father of computer science and contributed significantly to the WW2 effort, but his life came to a tragic end. Stefan Kopieczek explores his story.

Phil Trinh discovers how maths helps solve the mysteries of flight and love.

Liz Newton finds that having a small brain doesn't stop you doing great things.

José-Manuel Rey revisits a scene of the film A beautiful Mind.

Peter Macgregor explores the beautiful world of the infinite.

Josefina Alvarez describes the workings of the most famous search engine of them all. You'll need some linear algebra for this one, but it's worth the while!

The Arctic ice cap is melting fast and the consequences are grim. Mathematical modelling is key to predicting how much longer the ice will be around and assessing the impact of an ice free Arctic on the rest of the planet. Plus spoke to Peter Wadhams from the Polar Ocean Physics Group at the University of Cambridge to get a glimpse of the group's work.

Next year is a great one for biology. Not only will we celebrate 150 years since the publication of On the origin of species, but also 200 years since the birth of its author, Charles Darwin. At the heart of Darwin's theory of evolution lies a beautifully simple mathematical object: the evolutionary tree. In this article we look at how maths is used to reconstruct and understand it.

Lewis Dartnell turns the universe into a matrix to model traffic, forest fires and sprawling cities.

According to Darwin, natural selection is the driving force of evolution. It's a beautifully simple idea, but given the thousands of years that are involved, nobody has ever seen it in action. So how can we tell whether or not natural selection occurs and which of our traits are a result of it? In this article Charlotte Mulcare uses milk to show how maths and stats can provide genetic answers.

Bonuses are a fact of business life. Last year the Guardian newspaper calculated that the cash rewards paid to London's financial chiefs comfortably outstripped the UK's entire transport budget. With such large sums at stake, envy is bound to raise its ugly head, nver a good thing for company morale. So how should you decide who gets how much? Steven J. Brams suggests a method that's not only fair, but also encourages honesty.

Life is full of coincidences, but how do you work out if something is really as unlikely as it seems? In this article Rob Eastaway and John Haigh find chance in church and work out the odds.

NHS budgets, third world debt, predictions of global warming, inflation, Iraqi war dead, the decline of fish stocks or hedgehogs, the threat of cancer — there's hardly a subject people care about that comes without measurements, forecasts, rankings, statistics, targets, numbers of every variety. Do they illuminate or mislead? Introducing their new book, Michael Blastland and Andrew Dilnot take a look at numbers in the media and show that a little maths goes a long way in unravelling dodgy media claims.

Squares do it, triangles do it, even hexagons do it — but pentagons don't. They just won't fit together to tile a flat surface. So are there any tilings based on fiveness? Craig Kaplan takes us through the five-fold tiling problem and uncovers some interesting designs in the process.

Over the last few years the words string theory have nudged their way into public consciousness. It's a theory of everything in which everything's made of strings — or something like that. But why strings? What do they do? Where did the idea come from and why do we need such a theory? David Berman has an equation-free introduction for beginners.

In the fourth and final part of our series celebrating 300 years since Leonhard Euler's birth, we let Euler speak for himself. Chris Sangwin takes us through excerpts of Euler's algebra text book and finds that modern teaching could have something to learn from Euler's methods.

What's the risk of passive smoking? Or climate change? How big is the terrorist threat? And should we trust league tables? These issues concern all of us, but it's not always easy to make sense of the barrage of media information. David Spiegelhalter, Winton Professor for the Public Understanding of Risk, gives Plus his take on uncertainty.

How did we evolve our capacity for maths? Does maths piggy-back on our ability for language, or is it a completely separate faculty? Is it dependent on culture? Plus spoke to the cognitive psychologist Rosemary Varley to find some answers.

Phil Wilson continues our series on the life and work of Leonhard Euler, who would have turned 300 this year. This article looks at the calculus of variations and a mysterious law of nature that has caused some scientists to reach out for god.

John Napier was a clever man indeed. Besides inventing the logarithm, he developed ingenious calculating devices that fully exploit the power of the positional system. In this article Chris Sangwin tells you how to make your own set of Napier's bones and perform mathemagic with an interactive checker board.

Former Plus editor Helen Joyce explains how Plus made it big as a part of our series to celebrate Plus's tenth anniversary.

If you've ever redecorated a bathroom, you'll know that there are only so many ways in which you can tile a flat plane. But once you move into the curved world of hyperbolic geometry, possibilities become endless and the most amazing fractal structures ensue. Caroline Series and David Wright give a short introduction to the maths behind their beautiful images.

You might know the famous formula for an area of a circle, but why does this formula work? Tom Körner's explanation really is a piece of cake, served up with a hefty estimate of pi.

One of the many strange ideas from quantum mechanics is that space isn't continuous but consists of tiny chunks. Ordinary geometry is useless when it comes to dealing with such a space, but algebra makes it possible to come up with a model of spacetime that might do the trick. And it can all be tested by a satellite. Shahn Majid met up with Plus to explain.

Leonhard Euler, the most prolific mathematician of all time, would have celebrated his 300th birthday this year. In this article, the second in a four-part series on Euler and his work, Abigail Kirk explores one of the formulae that carry his name.

Plus celebrates its tenth birthday this year. Former editor and present executive editor of Plus, Robert Hunt, explores how maths popularisation in general, and Plus in particular, have changed over the last ten years.

Computer generated movies and electronic games: Joan Lasenby tells us about the mathematics and engineering behind them.

Plus went to see members of Norman Foster's group of architects to learn about the maths behind architecture.

If you've ever watched a flock of birds flying at dusk, or a school of fish reacting to a predator, you'll have been amazed by their perfectly choreographed moves. Yet, complex as this behaviour may seem, it's not all that hard to model it on a computer. Lewis Dartnell presents a hands-on guide for creating your own simulations — no previous experience necessary.

Leonhard Euler was one of the most prolific mathematicians of all time. This year marks the 300th anniversary of his birth. Robin Wilson starts off a four part series on Euler with a look at his life and work.

Plus magazine is celebrating its 10th birthday. To mark the occasion, the founding editors of Plus look back on the beginnings, see what has changed in maths and public understanding of maths and pick out some of the articles they liked best.

This issue of Plus is devoted to the winning entries of the Plus New Writers Award 2006.

Secondary school

General public

Winner Let 'em roll Dice are invaluable to many games, especially gambling games, but instead of playing with ordinary 1-6 numbered dice here are two interesting alternatives - with a twist! Runner-up Drinking coffee in the Klein Café Dusty books, chalky blackboards and checked shirts are all things usually associated with maths. But according to Jonathan Tims, pubs, hot chocolate and cats can be far more inspirational. Join him on a trip through shadow land. Runner-up The death of the lightning calculator Being good at mental arithmetic isn't going to gain you much street cred these days. But, as Owen Daniel explains, not so long ago it was a sure route to fame and fortune — even if you were a horse.

Winner An enormous theorem: the classification of finite simple groups Enormous is the right word: this theorem's proof spans over 10,000 pages in 500 journal articles and no-one today understands all its details. So what does the theorem say? Richard Elwes has a short and sweet introduction. Runner-up Damn lies "Lies, damned lies, and statistics..." Ben Parker tells us how to tell good statistics from bad, and make sure your cat is well-fed. Runner-up We must know, we will know Great minds spark controversy. This is something you'd expect to hear about a great philosopher or artist, but not about a mathematician. Get ready to bin your stereotypes as Rebecca Morris describes some controversial ideas of the great mathematician David Hilbert.

Learn about the aerodynamics of footballs and perfect your free kick.

What does a mathematician from the 3rd century BC have to do with tuning musical instruments in 17th century Europe? Benjamin Wardhaugh tells us about one of the more unusual places you might find Euclid's algorithm being used.

In the last article of this three-part series, Phil Wilson shows how simple graphs can tell you a lot about the economy — and not only in Slugworld.

You've probably seen pictures of the famed Mandelbrot set and its mysterious cousins, the Julia sets. In this article Robert L. Devaney explores the maths behind these beauties and shows that they're loaded with mathematical meaning.

What goes up must come down — or does it? Find out how to cheat gravity with Julian Havil.

When Kurt Gödel published his incompleteness theorem in 1931, the mathematical community was stunned: using maths he had proved that there are limits to what maths can prove. This put an end to the hope that all of maths could one day be unified in one elegant theory and had very real implications for computer science. John W Dawson describes Gödel's brilliant work and troubled life.

On the 25th of May 1997 a dramatic collision tore a hole into the space station Mir and sent it hurtling through space. As NASA astronaut Michael Foale tells Plus, the fate of Mir and its crew hinged on a classical set of equations.

In last issue's Graphical methods I Phil Wilson used maths to predict the outcome of a cold war in slug world. In this self-contained article he looks at slug world after the disaster: with only a few survivors and all infra-structure destroyed, which species will take root and how will they develop? Graphs can tell it all.

Groups are some of the most fundamental objects in maths. Take a system of interacting objects and strip it to the bone to see what makes it tick, and very often you're faced with a group. Colva Roney-Dougal takes us into their abstract world and puzzles over a game of Solitaire.

To arm or to disarm? This is the question in Phil Wilson's article, which explores the maths behind a cold war in slug world.

Everyone knows what symmetry is, and the ability to spot it seems to be hard-wired into our brains. Mario Livio explains how not only shapes, but also laws of nature can be symmetrical, and how this aids our understanding of the universe.

Get on a commuter train these days and you can virtually see people's brains crunching away at filling the numbers from 1 to 9 into a square grid. As the Sudoku craze shows no sign of slowing, Hardeep Aiden investigates its relatives and predecessors.

6174 is a very mysterious number. Yutaka Nishiyama explains why, and how beautiful mathematical oddities can inspire us to discover new mathematics.

Kurt Gödel, who would have celebrated his 100th birthday next year, showed in 1931 that the power of maths to explain the world is limited: his famous incompleteness theorem proves mathematically that maths cannot prove everything. Gregory Chaitin explains why he thinks that Gödel's incompleteness theorem is only the tip of the iceberg, and why mathematics is far too complex ever to be described by a single theory.

Carla Farsi is both an artist and a mathematician, who declared 2005 her Special Year for art and maths. Find out what she got up to, and what it's like being a part of both worlds.

Maths is not the first thing that springs to mind when you think about fighting crime. But a closer look reveals that it is behind many of the techniques that modern detectives rely on. Chris Budd investigates.

One hundred years ago, in 1905, Albert Einstein changed physics forever with his special theory of relativity. Since then his name — and hair do — have become synonymous with genius. John D Barrow looks at Einstein as a media star.

It's not that long ago that all you needed to run an airline was a few planes and some competent pilots. But now, with more of us zipping around the globe every year and the advent of no frills airlines, keeping an airline competitive has become a complicated business. Christine Currie explains how your airfare is calculated.

Most of us are aware that Einstein proved that everything was relative ... or something like that. But we go no further, believing that we aren't clever enough to understand what he did. Hardeep Aiden sets out to persuade readers that they too can understand an idea as elegantly simple as it was original.

What do computers and light switches have in common? Yutaka Nishiyama illuminates the connection between light bulbs, logic and binary arithmetic.

In the last issue Lewis Dartnell explained how chaos on the brain is not only unavoidable but also beneficial. Now he tells us why the same is true for our solar system and sends us on a journey that has been travelled by comets and spacecraft.

Physicist and cosmologist Paul Davies has made an unusual move into the infant discipline of astrobiology. He tells Plus about his interest in the big questions: what is life, how would we recognise aliens - and are they all around us?

According to Shakespeare, music is the food of love. But Jeffrey Rosenthal follows Galileo's observation that the entire universe is written in the language of mathematics - and that includes music.

In the second of two articles, Artur Ekert visits the strange subatomic world and investigates the possibility of unbreakable quantum cryptography.

Saying that someone is a chaotic thinker might seem like an insult - but, according to Lewis Dartnell, it could be that the mathematical phenomenon of chaos is a crucial part of what makes our brains work.

Tope Omitola looks back at the tragically short but inspiringly productive life of a true original: Evariste Galois.

The tsunami of December 26th 2004 has focused the world's attention on this terrifying consequence of an underwater earthquake. Michael McIntyre explores the underlying wave mathematics.

In the first of two articles, Artur Ekert takes a tour through the history of codes and the prospects for truly unbreakable quantum cryptography.

During the Second World War, the Allies' codebreakers worked at Bletchley Park to decipher the supposedly unbreakable Enigma code. Claire Ellis tells us about their heroic efforts, which historians believe shortened the war by two years.

Rachel Thomas looks at the life and work of pioneering woman mathematician Ada Lovelace, who foresaw computer-generated music and graphics, despite living long before the computer era.

Many people find no beauty and pleasure in maths - but, as Lewis Dartnell explains, our brains have evolved to take pleasure in rhythm, structure and pattern. Since these topics are fundamentally mathematical, it should be no surprise that mathematical methods can illuminate our aesthetic sense.

Did you know that you can't average averages? Or that Paris is rainier than London ... but it rains more in London than in Paris? Andrew Stickland explores the dangers that face the unwary when using a single number to summarise complex data.

Most of us have heard of "stealth" - a technology used by the military to disguise craft from enemy radar. But nature's stealth fighters are not so well known - creatures that use motion camouflaging to approach their prey undetected. Lewis Dartnell looks at the vector mathematics behind the phenomenon.

Mathematician and physicist John Baez declares himself fascinated by exceptions in mathematics. This interest has led him to study the octonions, and, through them, to find out more about the origins of complex numbers and quaternions. In the second of two articles, he talks about the characters of the different dimensions, beauty and utility in mathematics, and just why he likes dimension 8 so much.

Mathematician and physicist John Baez declares himself fascinated by exceptions in mathematics. This interest has led him to study the octonions, and, through them, to find out more about the origins of complex numbers and quaternions. In the first of two articles, he talks about connections between algebra and geometry, and the importance of lateral thinking in mathematics.

Frances Elwell looks at the eddies and currents, from the pungent problem of sewage outflow to the search for bodies of people who have fallen into rivers, explaining that fluid mechanics lies behind it all.

The three door problem has become a staple mathematical mindbender, but even if you know the answer, do you really understand it? Phil Wilson lets his imagination run riot in this intergalactic application of Bayes' Theorem.

Regular Plus contributor Lewis Dartnell reports on the scramble for million-dollar prizes that made mathematical headlines at the BA Festival of Science in September 2004.

Why do so many people say they hate mathematics, asks David Acheson? The truth, he says, is that most of them have never been anywhere near it, and that mathematicians could do more to change this perception - perhaps by emphasising the element of surprise that so often accompanies mathematics at its best.

As anyone starting out knows, the violin is a difficult instrument. It takes time before the novice player can expect to produce a musical note at the desired pitch, instead of a whistle, screech or graunch. Jim Woodhouse and Paul Galluzzo explain why.

Memory is fundamental to the way we think, and we use it in almost every activity. But most of us cannot imagine approaching the level of world record holder Hiroyuki Goto, who memorised and recited 42,195 digits of pi! Rob Eastaway asks if mere mortals can learn anything useful from such incredible feats of memory, and gives some hints on how to remember numbers.

How much evidence would you need before buying into a get rich quick scheme? Do high ice cream sales cause shark attacks? And just how likely was it that you were ever born? Andrew Stickland finds out that, when it comes to probability, our instincts can lead us seriously astray.

In issue 29 of Plus, we heard how a simple mathematical equation became the subject of a debate in the UK parliament. Chris Budd and Chris Sangwin continue the story of the mighty quadratic equation.

How does the uniform ball of cells that make up an embryo differentiate to create the dramatic patterns of a zebra or leopard? How come there are spotty animals with stripy tails, but no stripy animals with spotty tails? Lewis Dartnell solves these, and other, puzzles of animal patterning.

There are many different types of lottery around the world, but they all share a common aim: to make money. John Haigh explains why lotteries are the way they are.

It has often been observed that mathematics is astonishingly effective as a tool for understanding the universe. But, asks Phil Wilson, why should this be? Is mathematics a universal truth, and how would we tell?

In the early days of the UK National Lottery, it was quite common to see newspaper articles that looked back on what numbers had recently been drawn, and attempted to identify certain numbers as "due" or "hot". Few such articles appear now, and John Haigh thinks that perhaps the publicity surrounding the lottery has enhanced the nation's numeracy.

It isn't often that a mathematical equation makes the national press, far less popular radio, or most astonishingly of all, is the subject of a debate in the UK parliament. However, as Chris Budd and Chris Sangwin tell us, in 2003 the good old quadratic equation, which we all learned about in school, reached these dizzy pinnacles of fame.

Did you know that every instant, gravity waves from outer space are stretching and squeezing you - and everyone and everything else in the universe? Learning more about this mysterious radiation will help us to probe the structure and origins of the universe, explains Anita Barnes.

It is extraordinary to think that the diversity of the world we live in is based on a handful of elementary particles and a few fundamental forces. Peter Kalmus describes the combination of experimental and theoretical physics that has brought us to the understanding of today.

A biologist has developed a blood test for detecting a certain minor abnormality in infants. Obviously if you have blood samples from 100 children, you could find out which children are affected by running 100 separate tests. But mathematicians are never satisfied by the obvious answer. Keith Ball uses information theory to explain how to cut down the number of tests significantly, by pooling samples of blood.

Calculus is a collection of tools, such as differentiation and integration, for solving problems in mathematics which involve "rates of change" and "areas". In the second of two articles aimed specially at students meeting calculus for the first time, Chris Sangwin tells us how to move on from first principles to differentiation as we know and love it!

Following on from his article 'The prime number lottery' in last issue of Plus, Marcus du Sautoy continues his exploration of the greatest unsolved problem of mathematics: The Riemann Hypothesis.

In 1997 Garry Kasparov, then World Champion, lost an entire chess match to the IBM supercomputer Deep Blue, and it is only a matter of time before the machines become absolutely unbeatable. But the human brain, as Lewis Dartnell explains, is still able to put up a good fight by exploiting computers' weaknesses.

Calculus is a collection of tools, such as differentiation and integration, for solving problems in mathematics which involve "rates of change" and "areas". In the first of two articles aimed specially at students meeting calculus for the first time, Chris Sangwin tells us about these tools - without doubt, the some of the most important in all of mathematics.

Not only are paper models of geometric shapes beautiful and intriguing, but they also allow us to visualise and understand some important geometric constructions. Konrad Polthier tells us about the gentle art of paper folding.

Combinatorial Game Theory is a powerful tool for analysing mathematical games. Lewis Dartnell explains how the technique can be used to analyse games such as Twentyone and Nim, and even some chess endgames.

Marcus du Sautoy begins a two part exploration of the greatest unsolved problem of mathematics: The Riemann Hypothesis. In the first part, we find out how the German mathematician Gauss, aged only 15, discovered the dice that Nature used to chose the primes.

In the first of a new series 'Imaging Maths', Plus takes an illustrated tour of an extraordinary geometric construction: the Klein bottle.

The 2003 Dirac Lecturer, distinguished physicist Freeman Dyson, tells Plus why he is an optimist, what makes life interesting and why old-fashioned maths is what you need for physics.

The number chosen by the England captain for his Real Madrid shirt is rich in mysterious connotations. But mathematician Marcus du Sautoy backs a new theory to explain why Beckham has plumped for number 23.

All of science can be regarded as motivated by the search for rules behind the randomness of nature, and attempts to make prediction in the presence of uncertainty. Chris Budd describes the search for pattern and order in chaos.

The Riemann Hypothesis is probably the hardest unsolved problem in all of mathematics, and one of the most important. It has to do with prime numbers - the building blocks of arithmetic. Nick Mee, together with Sir Arthur C. Clarke, tells us about the patterns hiding inside numbers.

One million dollars is waiting to be won by anyone who can solve one of the grand mathematical challenges of the 21st century. In the second of two articles, Chris Budd looks at the well-posedness of the Navier-Stokes equations.

To study a system, mathematicians begin by identifying its most crucial elements, and try to describe them in simple mathematical terms. As Phil Wilson tells us, this simplification is the essence of mathematical modelling.

If you had a crystal ball that allowed you to see your future, what would you arrange differently about your finances? Plus talks to the Government Actuary, Chris Daykin about the pensions crisis, and how actuaries use statistical and modelling techniques to plan for all our futures.

One million dollars is waiting to be won by anyone who can solve one of the grand mathematical challenges of the 21st century. But be warned...these problems are hard. In the first of two articles, Chris Budd explains how to hit the bigtime.

Numbers are bandied around all the time in sports coverage - and cricket is particularly rich in statistics and rankings. It has probably not escaped your attention that the World Cup of cricket has just finished in South Africa (Australia won - again) and so to mark the occasion, Rob Eastaway tells Plus what it takes to be the best.

Some molecules - thalidomide, for example - come in both left and right handed versions, while others are indistinguishable from their reflections. Plus finds out about the role of mathematical symmetry in chemistry.

Imagine stepping inside your favourite painting, walking around the light-filled music room of Vermeer's "The Music Lesson" or exploring the chapel in the "Trinity" painted by Masaccio in the 15th century. Using the mathematics of perspective, researchers are now able to produce three-dimensional reconstructions of the scenes depicted in these works.

Currently, disabled computer users have a hard time inputting text, using laborious word-completion. Plus find out how this is changing, thanks to Dasher, a new open-source text-entry system based on arithmetic coding.

In 1694, a famous discussion between two of the leading scientists of the day - Isaac Newton and David Gregory - took place on the campus of Cambridge University. The discussion concerned the kissing problem, but it was to be another 260 years before the problem was finally solved.

How can a 3 hour long film like the Lord of the Rings fit on a single DVD? Hw cn U rd txt msgs? How do MP3s make music files smaller, so they can be downloaded faster off the Internet? All these things rely on the mathematics of data compression.

A new series of More or Less, BBC Radio 4's series devoted to all things numerical, starts on November 12th. Presenter Andrew Dilnot tells Plus about the motivation behind the programme.

When it comes to the science of the very small, strange things start happening, and our intuition ceases to be a useful guide. Plus finds out about the crazy quantum world, and spin that a politician would die for.

It was Euclid who first defined the Golden Ratio, and ever since people have been fascinated by its extraordinary properties. Find out if beauty is in the eye of the beholder, and how the Golden Ratio crosses from mathematics to the arts.

To make hard decisions, you need hard facts. Medical statistics can help us to decide what treatment to look for when we are ill, and to estimate our chances of recovery.

Today's digital world with its free flow of information, would not exist without cryptography to guarantee our privacy. Plus meets mathematician, author and broadcaster Simon Singh to find out about the science of secrecy.

In 1999 solicitor Sally Clark was found guilty of murdering her two baby sons. Highly flawed statistical arguments may have been crucial in securing her conviction. As her second appeal approaches, Plus looks at the case and finds out how courts deal with statistics.

Theoretical physicists are searching for a 'Theory of Everything' to reconcile quantum mechanics and relativity - the two great physical theories of the twentieth century. String theory is a current hot favourite, and some of the world's most eminent physicists tell us why.

What tactics should a soccer player use when taking a penalty kick? And what can the goalkeeper do to foil his plans? John Haigh uses Game Theory to find the answers, and looks at his World Cup predictions from last issue.

Fluid mechanics is the study of flows in both liquids and gases, and is therefore enormously important in understanding many natural phenomena, as well as in industrial applications. Geophysicist Herbert Huppert tells us what happens when two fluids of different densities meet, for example when volcanos erupt and hot ash-laden air is poured out into the atmosphere.

If your team scores first in a football match, how likely is it to win? And when is it worth committing a professional foul? John Haigh shows us how to use probability to answer these and other questions, and explains the implications for the rules of the game.

When we finally meet the Martians, John Conway believes they are going to want to talk mathematics. He talks to Plus about his Life game, artificial life and what we will have in common with extraterrestrials.

Clearly the modern electronic computer couldn't have been built before electronics existed, but it's not clear why computers powered by steam or clockwork weren't invented earlier. Tom Körner speculates on the historical reasons why computers were invented when they were.

Neuropsychologist Brian Butterworth tells us about research showing that even newborn babies have a basic understanding of number. It seems we are all mathematicians!

Chemists John Watling and Allen Thomas talk to Plus about the vital role of maths in presenting criminal evidence.

Infinite series occupy a central and important place in mathematics. C. J. Sangwin shows us how eighteenth-century mathematician Leonhard Euler solved one of the foremost infinite series problems of his day.

Paulus Gerdes takes us on a tour of the mathematical properties of some beautiful designs inspired by the traditional art of Angolan tribespeople.

This issue of Plus is a special, marking the occasion of Stephen Hawking's 60th birthday. Plus attended his Birthday Conference in Cambridge, where we interviewed some of the world's most influential mathematicians and physicists.

Plus is very proud to present Professor Stephen Hawking's own Birthday Symposium address.

Astronomer Royal Sir Martin Rees gives Plus a whistlestop tour of some of the more extraordinary features of our cosmos, and explains how lucky we are that the universe is the way it is.

Nobel Prizewinning Physicist Professor Gerardus 't Hooft has always been fascinated by the mathematical mysteries of nature. He tells Plus about his early life, and what our Universe might really be like.

Will we ever be able to make computers that think and feel? If not, why not? And what has all this got to do with tiles? Plus talks to Sir Roger Penrose about all this and more.

What happens when one black hole meets another? Professor Kip Thorne shows us how to eavesdrop on these cosmic events by watching for telltale gravitational waves.

Yes, you were right to wish you were in the other lane during this morning's commute! Nick Bostrom tells why we're usually caught in the slow lane.

As customers will tell you, overcrowding is a problem on trains. Fortunately, mathematical modelling techniques can help to analyse the changing demands on services through the day. Tim Gent explains.

Can you imagine objects that you can't measure? Not ones that don't exist, but real things that have no length or area or volume? It might sound weird, but they're out there. Andrew Davies gives us an introduction to Measure Theory.

During World Mathematical Year 2000 a sequence of posters were displayed month by month in the trains of the London Underground aiming to stimulate, fascinate - even infuriate passengers! Keith Moffatt tells us about three of the posters from the series.

The dangers of trading derivatives have been well-known ever since they were catapulted into the public eye by the spectacular losses of Nick Leeson and Barings Bank. John Dickson explains what derivatives are, and how they can be both risky, and used to reduce risk.

Sometimes a mathematical object can be so big that, however disorderly we make the object, areas of order are bound to emerge. Imre Leader looks at the colourful world of Ramsey Theory.

This pattern with kite-shaped tiles can be extended to cover any area, but however big we make it, the pattern never repeats itself. Alison Boyle investigates aperiodic tilings, which have had unexpected applications in describing new crystal structures.

Bill Casselman writes about the intriguing amateur mathematician Henry Perigal, who took his elegant proof of Pythagoras' Theorem literally to his grave - by having it carved on his tombstone.

Knots crop up all over the place, from tying a shoelace to molecular structure, but they are also elegant mathematical objects. Colin Adams asks when is a molecule knot a molecule? and what happens if you try to build a knot out of sticks?

Backgammon is said to be one of the oldest games in the world. In this article, Jochen Blath and Peter Mörters discuss one particularly interesting aspect of the game - the doubling cube. They show how a model using Brownian motion can help a player to decide when to double or accept a double.

A question which has been vexing astronomers for a long time is whether the forces of attraction between stars and galaxies will eventually result in the universe collapsing back into a single point, or whether it will expand forever with the distances between stars and galaxies growing ever larger. Toby O'Neil describes how the mathematical theory of dimension gives us a way of approaching the question.

Claude Shannon, who died on February 24, was the founder of Information Theory, which is the basis of modern telecommunications. Rachel Thomas looks at Shannon's life and works.

C. J. Budd and C. J. Sangwin show us how to create mazes, and explain why mazes and networks have much in common. In fact the study of mazes and labyrinths takes us into the dark territory of murder, suicide, adultery, passion, intrigue, religion and conquest...

Until you understand the basics of functions and algebra, the thought that a number can be predicted is a surprising one. And of course `magic' and `being surprised' are often the same thing. Rob Eastaway shows us how mathemagicians trade off the fact that you can usually predict precisely the outcome of doing something in mathematics, but only if you know the secret beforehand.

Over the past one hundred years, mathematics has been used to understand and predict the spread of diseases, relating important public-health questions to basic infection parameters. Matthew Keeling describes some of the mathematical developments that have improved our understanding and predictive ability.

Adam Smith is often thought of as the father of modern economics. In his book "An Inquiry into the Nature and Causes of the Wealth of Nations" Smith decribed the "invisible hand" mechanism by which he felt economic society operated. Modern game theory has much to add to Smith's description.

Arguably, the exponential function crops up more than any other when using mathematics to describe the physical world. In the second of two articles on physical phenomena which obey exponential laws, Ian Garbett discusses radioactive decay.

Steven J. Brams uses the Cuban missile crisis to illustrate the Theory of Moves, which is not just an abstract mathematical model but one that mirrors the real-life choices, and underlying thinking, of flesh-and-blood decision makers.

Last October, two mathematicians won £1m when it was revealed that they were the first to solve the Eternity jigsaw puzzle. It had taken them six months and a generous helping of mathematical analysis. Mark Wainwright meets the pair and finds out how they did it.

Why can't human beings walk as fast as they run? And why do we prefer to break into a run rather than walk above a certain speed? Using mathematical modelling, R. McNeill Alexander finds some answers.

Arguably, the exponential function crops up more than any other when using mathematics to describe the physical world. In the first of two articles on physical phenomena which obey exponential laws, Ian Garbett discusses light attenuation - the way in which light decreases in intensity as it passes through a medium.

Actuarial science began as the place where two branches of mathematics meet: compound interest and observed mortality statistics. Financial planning for the future is therefore rooted firmly in the past. John Webb takes us through some of the mathematics involved, introducing us to some of the colourful characters who led the way.

'Of the myriad strategems I employ to avoid useful work, the one I most enjoy is to envision how scientists of earlier eras would have made use of modern computers.' John L. Casti tells us how today's mathematicians are using computers to carry on the work of turn-of-the-century polymath d'Arcy Wentworth Thompson, who showed how mathematical functions could be applied to the shape of one organism to continuously transform it into other, physically similar organisms.

The harmonic series is far less widely known than the arithmetic and geometric series. However, it is linked to a good deal of fascinating mathematics, some challenging Olympiad problems, several surprising applications, and even a famous unsolved problem. John Webb applies some divergent thinking, taking in the weather, traffic flow and card shuffling along the way.

There are many errors that can occur when numbers are written, printed or transferred in any manner. Luckily, there are schemes in place to detect, and in some cases even correct, such errors almost immediately. Emily Dixon takes a break and discovers that codes are not just for sleuths.

Danielle Stretch looks back at the remarkable life of pioneering mathematician Emmy Amalie Noether (1882-1935). Despite her constant struggles to make her way in a man's world, she made significant contributions to the development of modern algebra.

We've all seen a traditional sundial, where a triangular wedge is used to cast a shadow onto a marked-out dial - but did you know that there is another kind? In this article, Chris Sangwin and Chris Budd tell us about a different kind of sundial, the analemmatic design, where you can use your own shadow to tell the time.

Those who understand compound interest are destined to collect it. Those who don't are doomed to pay it - or so says a well-known source of financial advice. But what is compound interest, and why is it so important? John H. Webb explains.

One of the most striking and powerful means of presenting numbers is completely ignored in the mathematics that is taught in schools, and it rarely makes an appearance in university courses. Yet the continued fraction is one of the most revealing representations of many numbers, sometimes containing extraordinary patterns and symmetries. John D. Barrow explains.

In the late 1940s, American painter Jackson Pollock dripped paint from a can on to vast canvases rolled out across the floor of his barn. Richard P. Taylor explains that Pollock's patterns are really fractals - the fingerprint of Nature.

Kevin Jones investigates the links between music and mathematics, throwing in limericks, Fibonacci and Scott Joplin along the way. Plus is proud to present an extended version of his winning entry for the THES/OUP 1999 Science Writing Prize.

Is the Universe finite, with an edge, or infinite, with no edges? Or is it even stranger: finite but with no edges? It sounds far-fetched but the mathematical theory of topology makes it possible, and nobody yet knows the truth. Janna Levin tells us more.

Underlying our vast global telecommunications networks are codes: formal schemes for representing information in machine-readable and transmissible formats. Kona Macphee examines the prefix property, one of the important features of a good code.

Robert Hunt concludes our Origins of Proof series by asking what a proof really is, and how we know that we've actually found one. One for the philosophers to ponder...

You might think that if you collected together a list of naturally-occurring numbers, then as many of them would start with a 1 as with any other digit, but you'd be quite wrong. Jon Walthoe explains why Benford's Law says otherwise, and why tax inspectors are taking an interest.

Images based on Lyapunov Exponent fractals are very striking. Andy Burbanks explains what Lyapunov Exponents are, what the much misunderstood phenomenon of chaos really is, and how you can iterate functions to produce marvellous images of chaos from simple mathematics.

Eugen Jost is a Swiss artist whose work is strongly influenced by mathematics. He sent us this Postcard from Italy, telling us about his work and the important roles that nature and numbers play in it.

For millennia, puzzles and paradoxes have forced mathematicians to continually rethink their ideas of what proofs actually are. Jon Walthoe explains the tricks involved and how great thinkers like Pythagoras, Newton and Gödel tackled the problems.

On 11th August 1999 a total eclipse of the Sun will be visible from parts of the UK. It will provide a spectacular display, but why is the Sun so interesting? Helen Mason explains.

At the Hewlett Packard campus in Bristol, a group of keen researchers are bringing together the worlds of advanced mathematics and fine art. Kona Macphee investigates.

We take reliable radio communications for granted, but accommodating many different users is not easy. Robert Leese explains how the mathematics of colouring graphs can help avoid interference on your mobile phone.

Johannes Kepler (1571-1630) is now chiefly remembered as a mathematical astronomer who discovered three laws that describe the motion of the planets. J.V. Field continues our series on the origins of proof with an examination of Kepler's astronomy.

In this article, we look at the physics behind the curved flight path of a returning boomerang, and explain that boomerangs are really a kind of gyroscope. We even show you how to bang up a boomerang yourself!

Here's how you can make your own cross-shaped boomerang - and it's safe enough to fly indoors! Hugh rolls up his sleeves and proves that theory isn't everything.

If boomerangs are really gyroscopes, then what are gyroscopes? In this article, we explore some more of the physics of gyroscopes, and demonstrate some interesting experiments you can do with them.

Whatever is so wonderful about point B that makes all the people at point A want to get there? Robert Hunt sits at point C, and muses on the problem.

Starting in this issue, PASS Maths is pleased to present a series of articles about proof and logical reasoning. In this article we give a brief introduction to deductive reasoning and take a look at one of the earliest known examples of mathematical proof.

Computer games and cinema special effects owe much of their realism to the study of fractals. Martin Turner takes you on a journey from the motion of a microscopic particle to the creation of imaginary moonscapes.

The term fractal, introduced in the mid 1970's by Benoit Mandelbrot, is now commonly used to describe this family of non-differentiable functions that are infinite in length. Find out more about their origins and history.

Practical problems often have no exact mathematical solution, and we have to resort to using unusual techniques to solve them. From navigation in the 17th century to postage stamps, see how this principle applies to a variety of real-life problems - and also learn how to use a piece of string to locate a German bomber!

Mike Yates looks at the life and work of wartime code-breaker Alan Turing. Find out what types of numbers we can't count and why there are limits on what can be achieved with Turing machines.

Quantum mechanics is the physics of the extremely small. With something so far outside our everyday experience it's not surprising to find mathematics at the heart of it all. But at the quantum scale nothing in life is certain... Peter Landshoff explains.

What is light? Sometimes it seems wave-like and sometimes particle like. See how Einstein applied his theory of relativity to the problem, predicted that photons have no mass and laid the foundations for quantum mechanics.

Coincidences are familiar to us all but what are the so-called laws of chance? From coin tossing to freak weather events, Geoffrey Grimmett explains how probability is at the heart of it all.

In the first of two articles, David Henwood discusses the vibrations that can be harnessed by musical instrument makers.

In his second article, David Henwood explains the role of mathematics in the design of Hi-Fi loudspeakers.

New technology has provided us with some amazing images - satellite images, medical images, even images beamed back from Mars. Julian Stander tells us about the increasing role of statistics in interpreting them.

Space probes, like NASA's recent Pathfinder mission to Mars, have radio transmitters of only a few watts, but have to transmit pictures and scientific data across hundreds of millions of miles without the information being completely swamped by noise. Read about how coding theory helps.

How do you choose a partner? Is it an irrational choice or is it made rationally, based on a mathematical model which analyses the best potential partner you are likely to meet?

The previous feature, "Mathematics, marriage and finding somewhere to eat" investigated the problem of finding the best potential partner from a fixed number of potential partners using a technique known as "optimal stopping". Inevitably, mathematicians and mathematical psychologists have constructed other models of the problem...

An account of how a prisoner of war's diary was recently decoded. Donald Hill wrote his diary in a numerical code, disguised as a set of mathematical tables, while in Hong Kong during and after the Japanese invasion of 1941.

Fibonacci, famous for the Fibonacci sequence, also introduced the decimal system into Europe.

Find out how modern telephone networks use mathematics to make it possible for a person to dial a friend in another country just as easily as if they were in the same street, or to read web pages that are on a computer in another continent.

The mathematics underlying today's complex telephone networks is still based on his work. Erlang was the first person to study the problem of telephone networks.

Here is an experiment that you can easily do yourself to test Bernoulli's equation. There are also 2 questions and answers.

The British General Election (May 1997) is an example of how simple mathematical ideas help in understanding information that involves numbers.

After 5,000 years, the game of Nine Men's Morris has succumbed to the power of modern computing, plus other recent mathematical discoveries in the world of games.

Have you ever been in an aeroplane on a smooth flight when suddenly the plane bumps up and down for a short time as it goes through turbulent air? The study of turbulence is used to understand a range of phenomena from the simple squirting of a jet of water to the activity of the sun.

Daniel Bernoulli (1700-1782) discovered the relationship between the density of a fluid in a pipe, the speed it is travelling in the pipe and the pressure exerted by the fluid against the walls of the pipe. This is the story of what happened.