Maths is weird. Numbers go on forever – and there are different kinds of forever. Prime numbers help cicadas survive. A (mathematical) ball can be cut up and then put back together, without any gaps, to make a ball twice the size, or a million times the size, of the original. There are shapes that have fractional dimensions and curves that fill a plane leaving no holes. While bored by a dull presentation, physicist Stanislaw Ulam wrote out numbers, starting from 0, in a spiral form, marked in all the prime numbers, and found that many primes lie on long diagonals – a fact still not fully explained.
We forget sometimes how weird maths is because we’re so used to dealing with what seem like ordinary numbers and calculations, the stuff we learn about in school or use every day. Yet the fact that our brains are so adept at thinking mathematically, and, if we choose, at doing really complex and abstract maths, is surprising. After all, our ancestors, tens or hundreds of thousands of years ago, didn’t need to solve differential equations or dabble in abstract algebra in order to stay alive long enough to pass on their genes to the next generation. While searching for their next meal or a place to shelter, there was nothing to be gained from musing about geometry in higher dimensions or theories of prime numbers. Yet we’re born with brains that have the potential to do these things, and to uncover, with each passing year, more and more extraordinary truths about the mathematical universe. Evolution has provided us with this skill: but how and why? Why are we, as a species, so good at doing something that has every appearance of being just an intellectual game?
Somehow maths is woven into the very fabric of reality. Dig deep enough and we find that what seemed to be tangible bits of matter or energy – electrons or photons, for instance – dissolve into immateriality, becoming mere waves of probability, and all we’re left with is a ghostly calling card in the form of some intricate but beautiful set of equations. In some sense, mathematics underpins the physical world around us, forming an invisible infrastructure. Yet it also goes beyond this, into abstract realms of possibility that may forever remain purely exercises of the mind.
We’ve chosen in this book to highlight some of the more extraordinary and fascinating areas of maths, including those where exciting new developments are in the offing. In some cases, they have links with science and technology – particle physics, cosmology, quantum computers, and the like. In others, they represent, for now at least, maths for maths sake, and are adventures into an unfamiliar land that exists only in the mind’s eye. We’ve chosen not to shy away from certain subjects just because they’re hard. One of the challenges in describing many aspects of maths for a general audience is that they’re far removed from everyday experience. But, in the end, some way can always be found to link what today’s explorers and pioneers at the frontiers of mathematics are doing with the world of the familiar, even if the language we have to use isn’t as precise as academics would ideally choose. It’s perhaps true to say that if something, however obscure, can’t be explained reasonably well to a person of normal intelligence then the explainer needs to improve their understanding!
This book came about in an unusual way. One of us (David) has been a science writer for more than 35 years and has written many books on astronomy, cosmology, physics, and philosophy, and even an encyclopaedia of recreational maths. The other (Agnijo) is a brilliant young mathematician and child genius, with an IQ of at least 162 according to Mensa, who, at the time of writing, has just finished training in Hungary in preparation for the 2017 International Mathematics Olympiad. Agnijo started coming to David for tuition in maths and science at the age of 12. Three years later, we decided to write a book together.
We sat down and brainstormed the topics we wanted to cover. David, for instance, came up with higher dimensions, the philosophy of maths, and the maths of music, while Agnijo was keen to write about large numbers (his personal passion), computation, and the mysteries of primes. Right from the start we chose to lean towards anything unusual or downright weird and to connect this weird maths, where possible, with real-world issues and everyday experience. We also made a commitment not to shy away from subjects just because they were tough, adopting as a mantra that if you can’t explain something in plain language then you don’t properly understand it. David generally took on the historical, philosophical, and anecdotal aspects of each chapter while Agnijo grappled with the more technical aspects. Agnijo fact-checked David’s work, and David combined all the writing into finished chapters. It all worked surprising well! We hope you enjoy the result.