Eugenia Cheng - Beyond Infinity: An Expedition to the Outer Limits of the Mathematical Universe

BEYOND INFINITY

Eugenia Cheng is Honorary Fellow in Pure Mathematics at the University of Sheffield and Scientist in Residence at the School of the Art Institute of Chicago. She was educated at the University of Cambridge and did post-doctoral work at the Universities of Cambridge, Chicago and Nice.

Since 2007 her YouTube lectures and videos have been viewed over a million times. A concert pianist, she also speaks French, English and Cantonese, and her mission in life is to rid the world of maths phobia. She is the author of How to Bake Pi.

Also by Eugenia Cheng

How to Bake Pi: Easy recipes for understanding complex maths

First published in Great Britain in 2017 by

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Copyright Eugenia Cheng, 2017

The moral right of the author has been asserted.

All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the publisher of this book.

A CIP catalogue record for this book is available from the British Library.

eISBN 978 178283 0818

In memory of Sara Al-Bader

who taught me by example that infinite love can fit into a finite life.

Contents

Prologue

I hate airports.

I find airports stressful, crowded, noisy. There are usually too many people, too many queues, not enough seats, and unhealthy food everywhere tempting me to eat it. Its a shame when this is the way travelling starts, as it makes me dread the journey. Travelling should be an exciting process of discovery. Airports along with cramped economy seating too often mar what is the almost miraculous and magical process of flying somewhere in a plane.

Mathematics should also be an exciting process of discovery, an almost miraculous and magical journey. But it is too often marred by the way it starts, with too many facts or formulae being thrown at you, and stressful tests and unpalatable problems to solve.

By contrast, I love boat trips.

I love being out on the open water, feeling the wind in my face, watching civilisation and the coastline recede into the distance. I like heading towards the horizon without it ever getting any closer. I like feeling some of the power of nature without being entirely at its mercy: Im not a sailor, so usually someone else is in charge of the boat. Occasionally there are boats I can manage, and then the exertion is part of the reward: a little rowing boat that I once rowed around a small moat encircling a tiny chateau in France; a pedal boat along the canals of Amsterdam; punting on the river Cam, although after once falling in I was put off for life, just like some people are put off mathematics for life by bad early experiences. I have taken boat trips to see magnificent whales off the coast of Sydney and Los Angeles, or seals and other wildlife off the coast of Wales. Then there are the ferries crossing the Channel to France that started most of our family holidays when I was little, before the improbable Eurostar was built. How quickly we humans can come to take something for granted even though it previously seemed impossible!

These days I rarely take boats with the purpose of getting anywhere rather, the purpose is to have fun, see some sights or some nature, and possibly exert myself. The one exception is the Thames River Ferry, which is a very satisfying way to commute into Central London, joyfully combining the fun of a boat trip and a journey with a destination.

I love abstract mathematics in somewhat the same way that I love boat trips. Its not just about getting to a destination for me. Its about the fun, the mental exertion, communing with mathematical nature and seeing the mathematical sights. This book is a journey into the mysterious and fantastic world of infinity and beyond. The sights well see are mind-boggling, breathtaking and sometimes unbelievable. We will revel in the power of mathematics without being at its mercy, and we will head towards the horizon of human thinking without that horizon ever getting any closer.

part one

THE JOURNEY

What Is Infinity?

I nfinity is a Loch Ness Monster, capturing the imagination with its awe-inspiring size but elusive nature. Infinity is a dream, a vast fantasy world of endless time and space. Infinity is a dark forest with unexpected creatures, tangled thickets and sudden rays of light breaking through. Infinity is a loop that springs open to reveal an endless spiral.

Our lives are finite, our brains are finite, our world is finite, but still we get glimpses of infinity around us. I grew up in a house with a fireplace and chimney in the middle, with all the rooms connected in a circle around it. This meant that my sister and I could chase each other round and round in circles forever, and it felt as if we had an infinite house. Loops make infinitely long journeys possible in a finite space, and they are used for racetracks and particle colliders, not just children chasing each other.

Later my mother taught me how to program on a Spectrum computer. I still smile involuntarily when I think about my favourite program:

10 PRINT HELLO

20 GOTO 10

This makes an endless loop an abstract one rather than a physical one. I would hit RUN and feel delirious excitement at watching HELLO scroll down the screen, knowing it would keep going forever unless I stopped it. I was the kind of child who was not easily bored, so I could do this every day without ever feeling the urge to write more useful programs. Unfortunately this meant my programming skills never really developed; infinite patience has strange rewards.

The abstract loop of my tiny but vast program is made by the program going back on itself, and self-reference gives us other glimpses of infinity. Fractals are shapes built from copies of themselves, so if you zoom in on them they keep looking the same. For this to work, the detail has to keep going on forever, whatever that means certainly beyond what we can draw and beyond what our eye can see. Here are the first few stages of some fractal trees and the famous Sierpinski triangle.

If you point two mirrors at each other you see not just your reflection, but the reflection of your reflection, and so on for as long as the angle of the mirrors permits. The reflections inside the reflections get smaller and smaller as they go on, and in theory they could go on forever like the fractals.

We get glimpses of infinity from loops and self-reference, but also from things getting smaller and smaller like the reflections in the mirror. Children might try to make their piece of cake last forever by only ever eating half of whats left. Or perhaps youre sharing cake, and everyone is too polite to have the last bite so they just keep taking half of whatevers left. Im told that this has a name in Japanese: