Richard Elwes - Chaotic Fishponds and Mirror Universes: The Strange Maths Behind the Modern World

Richard Elwes

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First published in 2013

Copyright Richard Elwes 2013

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eBook ISBN: 978 1 78087 561 3

UK and associated territories: Print ISBN 978 1 78087 160 8

Illustrations by Patrick Nugent

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Dr Richard Elwes is a writer, teacher and researcher in Mathematics and a Visiting Fellow at the University of Leeds. He contributes to New Scientist and Plus Magazine and publishes research on model theory. Dr Elwes is a committed popularizer of mathematics, which he regularly promotes at public lectures and on radio.

The mathematical hierarchies of knowledge

Mathematical definitions of the typical

Chemical graph theory and the make-up of molecules

The backbone of the computer age

Projective geometry and the world of art

The mathematics of digital photography

The mathematics of planetary motion

The mathematics of machine learning

The mathematics of DNA

The mathematics of elections

Triangulation and computer-generated imagery

The (a) symmetries of the universe

The mathematics of group-testing

The untidy growth of populations

The mathematical basis of decision theory

The shapes of the universe

The mathematics of weather forecasting

Statistical illusions and numerical mirages

GPS geometry and Einsteins explorations

Optimizing the world

The mathematics of social networks

The mathematics of significance

Fixed points and the nature of equilibrium

The mathematics of space travel

Futures, options and the ups and downs of markets

The tricky world of timetables

The extraordinarily useful geometry of optics

Mathematical models and the spread of infection

The mathematics of sound and light

The algorithm behind Googles PageRank

The mathematics of queues

Correcting mistakes in digital communication

The mathematics of robot movement

The mathematics of energy and entropy

Type theory and programming

Of all the subjects studied, debated and fought over in the course of human history, I happen to believe that the most fascinating is mathematics. Thats a bold claim perhaps mystifying to readers who were bored or baffled by the subject at school. Well, of course fascination is in the eye of the beholder, and certainly there will be those who need some persuading. I hope this book will go some way towards doing that.

What is irrefutable, however, is that in modern life mathematics is both important and ever-present. Even the most entrenched maths-hater has an awareness that it plays a central role in todays world, touching our lives in more ways than ever before. But that is where the details are liable to become hazy yes, important, but where exactly is it used, and in what ways?

In response, I present in the pages that follow a selection of 35 diverse applications of mathematics. I attempt to unravel some of the principles that underlie aspects of our daily lives, as well as those that inform todays boldest thinkers. We will touch on numerous branches of science, seeing how chaos theory can illuminate our understanding of population growth and why the universes black holes are contrary to all expectations rich in information.

Money that universal experience of numbers is explored in the context of options and futures, along with the gamblers intimate (if rocky) relationship with probability. The computing revolution, of course, suffuses the discussion, as does its offspring the Information Age. Most people have a sense that mathematics is deeply involved in these developments the air is thick with talk of binary code and that now popular word, the algorithm. From delving into the innards of search engines, to investigating the structures of social networks and illuminating the ideas behind programming languages and CGI imagery, the aim is to bring this mathematics out of the shadows.

I want to show how mathematics can be used to solve some profoundly pragmatic problems, helping businesses to run efficiently, highlighting flaws in our democracies, providing valuable ammunition in the fight against disease, and analysing the highs and lows of our economic systems. It can even shine a light on the foibles of human psychology, demonstrating where we are prone to make decisions against our best interests. We will discover ways to tell fact from fiction too, by way of those much-used and much-abused numbers: statistics.

All this extremely useful mathematics does much to free the modern mathematician from the ivory tower he, or she, is meant to inhabit. This is not to say that todays mathematicians have lost their imagination or no longer reach for the stars. Indeed, in 1969 Apollo 11s touchdown on the moon was made possible by the analysis of one particularly fiendish mathematical problem. In fact, the stars and planets have provided us with numerous mathematical insights over the centuries, which can now be turned upon the still bigger questions about the nature of matter and the universe.

I hope that, by the end of this book, readers will have a more precise sense of where mathematics fits into modern life and, en route, that some doubters become devotees of the subject that I find so endlessly, gloriously, fascinating.

The mathematical hierarchies of knowledge

Imagine a scene. Sarah and Sam used to be a couple. Today they are meeting again, for the first time since their relationship broke down. Since then, Sarah has found a new boyfriend, but has decided not to tell Sam. Unbeknownst to her, however, Sam already does know, because a mutual friend, Saul, spotted the two together and told Sam.

It sounds as though we have left mathematics for the world of romantic comedy or soap-opera. In fact, the dramas of human relationships provide examples where mathematicians concepts of different orders of knowledge come into play. These ideas are central to the mathematical genre of game theory and in philosophy. Their practical implications continue to be debated, especially in economics. In a competitive market, possessing the same (or superior) knowledge to ones rivals can make all the difference.