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Many of the deepest and most important areas of mathematics have emerged from questions about extremes—the shortest path between two points on a curved surface, the smallest area spanning a wire, or the fewest colors needed to make a map. Mathematicians have been pushing restlessly toward extremes for thousands of years. The isoperimetric problem, for example—which asks for the shortest route enclosing a given area—can be traced to ancient Carthage. By contrast, it was only in 2017 that the densest ways to pack identical spheres into a 24-dimensional space was proven. In Reaching for the Extreme, bestselling author Ian Stewart, one of the world’s most popular writers on mathematics, presents a dazzling, wide-ranging tour of math’s outer limits.

What was your motivation for writing this book?

Ian Stewart: All of my popular math books share a common motivation. Math is hugely important, beautiful, challenging, fascinating, and it underpins today’s world. But these facts aren’t as widely understood as they should be. The math we study at school is just one tiny part of a much broader enterprise, and it’s not really typical of the subject at more advanced levels. I’ve always wanted to show readers what math is really like, challenge some of the common myths, and enjoy the subject along the way. I’m not so much ‘making’ math interesting or fun, as revealing the interest and fun that are already present. 

The book actually started with a different idea: to discuss ‘minimal surfaces’, with applications like soap bubbles. Soap films adopt whichever shape has the smallest area, subject to physical constraints such as enclosing a given volume of air. That’s why bubbles are spherical (if nothing disturbs them). A surprising amount of deep mathematics has arisen from problems of this kind, and it applies to a lot more than just bubbles.

However, this topic is very specialized, so my editors suggested broadening the scope to a general exploration of the role of extremes in math. Not just ‘optimization’’ — finding the best way to achieve some goal — but more imaginative ideas united by the common theme of pushing the boundaries of mathematics as far as they can possibly go.

The main problem after that wasn’t finding suitable topics: it was deciding what to include and what to omit. I usually write books that start out too long, and throw bits away. At least two chapters bit the dust.

Can you give us some simple examples?

IS: My story about three, four, or five ‘items’ is an exact description of the famous four color problem (Fewest Colors). The items are colors, and the problem is to color any map so that regions that share a common boundary have different colors. It’s easy to see that three colors aren’t enough, and a century or so ago it was proved that five always do the job. But it took a heroic effort, with computer assistance, to close the gap and prove that four colors suffice.

A very different example is the idea of a geodesic (Shortest Path): a curve on a surface that joins two given points and is the shortest curve that does so. If the surface is flat, the answer is simple: a straight line. But what if it’s curved?

Different again is the Kepler conjecture (Tightest Pack) , which dates back to 1611. Which arrangement of identical spheres in space packs them as closely together as possible? Johannes Kepler thought he knew the answer, over four centuries ago, and he was right — but a proof of that was lacking until 1998.

Other examples are the quest for the biggest possible number and the smallest possible number. Those probably sound silly, because neither of those things exists. But historically those questions led to deep and important concepts of infinity and infinitesimals; then to fierce controversies about those ideas; then to a clever but indirect way to get around the objections… and finally to a logically acceptable way to confront them head-on and use them to advantage.

What uses does that kind of mathematics have? What has it done for humanity in general?

IS: Most applications of math are hidden behind the scenes. This is entirely sensible: no one wants to have to learn spherical trigonometry to use their car’s satellite navigation. So the math is squirreled away inside electronic circuits and software. But, in a way, that’s a pity, because no one realizes the math is there. In fact, satellite navigation relies on at least eight different areas of math, including celestial mechanics, Einstein’s theories of special and general relativity — even number theory.

Some math is discovered as a direct result of specific real-world problems; Newton invented calculus to understand motion; especially how the planets move. Calculus originally depended on the vexed issue of infinitesimals, until ways were found to avoid them, so infinitesimals represent an idea with profound effects on everyday life.

That said, many applications arise when an area of math invented for totally different reasons — including ‘that looks interesting’ — suddenly turns out to be vital to some area of human activity. Minimal surfaces have applications to biology (the iridescent colors on beetles’ wings), to nanotechnology (manufacturing very small substances and devices), and to cosmology (the nature of the universe and our place within it). Sphere packing has connections to error-correcting codes in digital communications. Geodesics govern the paths aircraft fly along and the fundamental physics of gravity. Graph theory finds the shortest route between two cities, or between a depot and delivery addresses. And the math behind AI has led to huge progress on how proteins fold, which is important in medicine.

I don’t know of any important practical applications of the four color theorem; I do know that if one ever arises, it won’t be remotely related to maps. But its story is fascinating because the problem sounds very simple, but the answer (so far!) has turned out to be extremely complicated.

Does your book have any other main themes?

IS: Aside from extremes, the other main theme is ‘how mathematicians think’. Where do mathematicians get their problems from? How do they go about solving them? Where do they get new ideas from? Do they work in isolation, compete, or cooperate? Why do they seek proofs of statements that seem obvious? Why do many of them devote time and energy to ‘useless’ areas such as number theory, instead of tackling ‘practical’ problems head on? What, for that matter, is mathematics? What are mathematicians?

Standard answers to the last two questions are that math is ‘what mathematicians do’, and mathematicians are ‘people who do mathematics’. Not terribly helpful. But consider the same questions for business. What is business? What business people do. What is a business person? Someone who does business. My view is that in both cases a vital point is being missed. A business person doesn’t just do business: they see opportunities to do it where most of us don’t. Similarly, a mathematician is a person who sees opportunities to do math where most of us don’t. Mathematicians are immersed in a long tradition of seeking such opportunities, and collectively they’ve amassed a huge toolkit of methods for exploiting them. So in each chapter I try to explain where the problems came from, how mathematicians tried to solve them (mistakes and all, sometimes), which different approaches were developed, and — as best I can reconstruct — what was going through their minds when they worked on the problem.

The book consists of separate, stand-alone chapters. Why did you choose to write it that way?

IS: I like diversity. Math is such a wide-ranging subject, far broader than most of us imagine. Roughly a million pages of new math is produced every year by the world’s research mathematicians. Not just more complicated sums: new concepts new theories, entire new areas from time to time.

Sometimes it’s possible to write a whole book as a single story. A novel. There are some very good books that discuss a single major idea: Fermat’s last theorem, the four color problem, the Kepler conjecture, prime numbers, pi. But once I was persuaded to abandon the bubbles idea, the more general themes of the book demanded not a novel, but a collection of short stories. Linked by two common themes: extremes, and how mathematicians think.

There’s one other advantage. Math is a demanding subject. You need to get your brain in gear. Breaking the book into bite-sized pieces makes the experience more pleasant. I always try my best to keep the material accessible to anyone interested enough to read it. A popular math book isn’t a research text. In this book, if one chapter seems difficult, you can always skip the rest of it and go to the next chapter on a different topic.

I do advocate returning to the difficult bit later and having another go, if you feel so inclined, but this kind of book should be a pleasurable experience. No one’s going to grade you on how much of it you’ve read.

A few topics are more challenging than the others. What led you to include those?

IS: I have a tendency to push my luck sometimes. But there’s a reason. Readers of popular science and math often want more than easy entertainment. They really do want to get to grips with the topic. Not, as I’ve just said, by passing tests. But if too much is left out, all we get is ‘dumbing down’. I prefer ‘dumbing up’, and the way to do that is to have a few pages that push the boundaries.

Even then, I try to explain the ideas as accessibly as I can, with analogies and simple examples. But as long as you realize that some of the technical terms (‘infinite-dimensional module’, say) are included only as labels for something too complicated to explain, it’s possible to follow the story. We do this all the time with fiction: no one asks for Elizabeth Bennet’s entire family tree or the precise layout of Mr. Darcy’s house.

The most challenging chapter in this book is, I’m pretty sure, Chapter 14: ‘weirdest symmetry’. Even professional mathematicians have difficulties with the technical side of this topic. Only a few amazingly clever specialists really understand it. However, the underlying story is so astonishing that I was determined to keep it in. And symmetry is absolutely fundamental in math, as well as being pretty.

The story in Chapter 14 l began with a strange coincidence: a number that arose independently in two radically different areas of math. The number was 196,884. One area was group theory, the study of symmetry in its most abstract and general form. The other was number theory, the topic of ‘modular forms’.

A few mathematicians felt that the same strange number turning up in two different areas was a clue. But to what? As it turned out, the gigantic ‘monster’ simple group and an idea so mad that its inventors dubbed it ‘monstrous moonshine’. It turned out to have mysterious links to quantum field theory, which nobody expected.

You tell a lot of historical stories. How important is the history of mathematics?

IS: Math makes a lot more sense if you have some idea of its history. Not in the technical detail that a professional historian would demand, however. Most math goes through a long process of development, from early vague ideas through formal definitions, from unproved conjectures to proofs or disproofs. Notation and viewpoint can change, to such an extent that even an expert can find it hard to decipher exactly what was intended fifty years ago.

What matters is the broad narrative of the origins and growth of the area; its main concepts and results. Those provide a context for the math, based on ideas rather than formal definitions and theorems. The context can reveal links between apparently distinct areas.

Historical details also add a personal element. We learn not just about the math, but the people who discovered or invented it. What their lives and times were like. This offers interesting perspectives on their thinking. Also, an occasional historical story helps to ease the flow of the narrative. However, you don’t need to be told what they ate for breakfast. The personal approach can be overdone.

Haven’t computers and AI made mathematics obsolete?

IS: No.

I’d like to leave it at that, but perhaps a bit more is needed. First, both computers and AI emerged from enormous quantities of math, developed over centuries. The logic underlying computer electronics goes back to ancient Greece and the work of George Boole in 1854. The possibility of AI was envisioned by mathematicians Augusta Ada King (Countess of Lovelace) in 1843 and Alan Turing in 1950. Its underlying techniques rest heavily on math, especially linear algebra (matrices) and probability theory. Of course, massive advances in engineering, physics, chemistry, and other subjects were also required. But future advances in AI will need all kinds of math, old and new.

Computers, and AI are tools. Like all powerful tools, they render some previous methods obsolete. More importantly, they open up new ones. Computers and AI no more render math and mathematicians  obsolete than the microscope rendered biology and biologists obsolete. They’re already changing how we go about doing math, but that’s all to the good.

Maybe in future AI really will make math obsolete. But only by making human beings obsolete. That’s a different issue, currently causing a great deal of concern.

What are the main messages you’d like readers to take from your book?

IS: There’s much more to math than what you were taught by that name at school.

The answers aren’t all in the back of the textbook. Math is a thriving research area, with many new discoveries and ideas happening all the time.

You yourself may not be conscious that you’re using math, but if the math in today’s world were suddenly taken away, everything would collapse. A 2024 survey revealed that one in six UK workers were in mathematical science occupations, generating one fifth of the country’s gross value added: $660 billion. The population of the USA is five times as big: do the math.

Math isn’t just dull ‘sums’. It’s beautiful, wonderful, and full of surprises. You don’t need to be an expert to appreciate those things.

Ian Stewart is an award-winning mathematician and bestselling author of many popular math books, including Professor Stewart’s Cabinet of Mathematical Curiosities, Do Dice Play God?, Significant Figures, Calculating the Cosmos, In Pursuit of the Unknown, and Professor Stewart’s Casebook of Mathematical Mysteries. He is professor emeritus of mathematics at the University of Warwick and a fellow of the Royal Society.