What do you think?
Rate this book
Reaching for the Extreme: How the Quest for the Biggest, Fewest, and Weirdest Makes Math
From bestselling author and mathematician Ian Stewart, the fascinating story of the extreme problems that have driven math forward from antiquity to today
"Stewart has a genius for explanation."—New Scientist
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.
Stewart tells the stories of sixteen superlative problems—their history, the struggles to solve them, and the uses of some of the results. From the biggest number to the smallest, the fastest fall to the weirdest symmetry, and the best fold to the shortest proof, these questions are either pure thought experiments or are motivated by real-world challenges. The Plateau problem, about the geometry of soap bubbles, led to the notion of a minimal surface—now used in cosmology, biology, and other fields. Meanwhile, the 2023 discovery of a single tile shape that covers the infinite plane without repeating the same pattern has no application—yet.
Reaching for the Extreme illuminates how mathematicians drive knowledge forward by reaching for the edges and solving some of the world’s most fascinating problems.
"Stewart has a genius for explanation."—New Scientist
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.
Stewart tells the stories of sixteen superlative problems—their history, the struggles to solve them, and the uses of some of the results. From the biggest number to the smallest, the fastest fall to the weirdest symmetry, and the best fold to the shortest proof, these questions are either pure thought experiments or are motivated by real-world challenges. The Plateau problem, about the geometry of soap bubbles, led to the notion of a minimal surface—now used in cosmology, biology, and other fields. Meanwhile, the 2023 discovery of a single tile shape that covers the infinite plane without repeating the same pattern has no application—yet.
Reaching for the Extreme illuminates how mathematicians drive knowledge forward by reaching for the edges and solving some of the world’s most fascinating problems.
- GenresMathematics
352 pages, Hardcover
First published February 12, 2026
15 people are currently reading
90 people want to read
About the author
Ian Stewart
274 books762 followersIan Nicholas Stewart is an Emeritus Professor and Digital Media Fellow in the Mathematics Department at Warwick University, with special responsibility for public awareness of mathematics and science. He is best known for his popular science writing on mathematical themes.
--from the author's website
Librarian Note: There is more than one author in the GoodReads database with this name. See other authors with similar names.
--from the author's website
Librarian Note: There is more than one author in the GoodReads database with this name. See other authors with similar names.
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 of 1 review
February 12, 2026
Ian Stewart is arguably the UK's best raconteur of mathematics - here he takes on some of the extremes of the mathematical world, and in doing so gives us some real insights into what makes mathematicians tick.
There's a good mix here of the flashy fun aspects of maths - think, for instance of the wonders of infinity or the monster group - and the solid everyday that nonetheless can turn up surprises. The book is littered with little insights. For example, if we think it's easy to work out the area of a rectangle by dividing it up into unit squares, what do you do with one that measures square root of two by pi?
You'll find yourself jumping around from what lies beneath calculus to game theory (rock, paper, scissors anyone? - I hadn't realised a version of this game dates back around 2,000 years). One minute you'll be considering colouring maps and the next finding the shortest distance between two points on a curved surface.
Some of the mathematics here has everyday applications, some can be used in fairly abstruse corners of science and some is pure mathematical puzzle solving with no obvious application ever. You pays your money and you takes your choice.
There were a few points where Stewart lost me. Some of these were simply due to going on too far. I enjoyed the tiling chapter to begin with, but as we got onto more and more obscure aperiodic tiles there was a bit of a feeling of 'we get the point, move on'. In other cases, Stewart had the problem of dealing with mathematics that is well beyond the grasp of a general reader like me. He is clearly aware of this, but rather than avoid it, makes the discussion so vague that it's difficult to feel you are getting anywhere.
This was particularly the case in the 'weirdest symmetry' chapter, which brings in the aforementioned monster group. There's some nice material on what groups are, but when we get onto that monster, the 'largest sporadic simple group', it's way beyond comprehension, when we're dealing with something it's was difficult even to prove whether or not it existed. It doesn't really help that Stewart then ties this into string theory, an aspect of physics that many believe is more mathematical play than anything connected to physical reality, which has been notably described as 'not even wrong'.
There's a lot to like, though, in this generous meander through eighteen chapters of different mathematical pondering. It'll certainly stretch your mind - and if occasional it leaves us mere mortals behind, that's not always a bad thing.
There's a good mix here of the flashy fun aspects of maths - think, for instance of the wonders of infinity or the monster group - and the solid everyday that nonetheless can turn up surprises. The book is littered with little insights. For example, if we think it's easy to work out the area of a rectangle by dividing it up into unit squares, what do you do with one that measures square root of two by pi?
You'll find yourself jumping around from what lies beneath calculus to game theory (rock, paper, scissors anyone? - I hadn't realised a version of this game dates back around 2,000 years). One minute you'll be considering colouring maps and the next finding the shortest distance between two points on a curved surface.
Some of the mathematics here has everyday applications, some can be used in fairly abstruse corners of science and some is pure mathematical puzzle solving with no obvious application ever. You pays your money and you takes your choice.
There were a few points where Stewart lost me. Some of these were simply due to going on too far. I enjoyed the tiling chapter to begin with, but as we got onto more and more obscure aperiodic tiles there was a bit of a feeling of 'we get the point, move on'. In other cases, Stewart had the problem of dealing with mathematics that is well beyond the grasp of a general reader like me. He is clearly aware of this, but rather than avoid it, makes the discussion so vague that it's difficult to feel you are getting anywhere.
This was particularly the case in the 'weirdest symmetry' chapter, which brings in the aforementioned monster group. There's some nice material on what groups are, but when we get onto that monster, the 'largest sporadic simple group', it's way beyond comprehension, when we're dealing with something it's was difficult even to prove whether or not it existed. It doesn't really help that Stewart then ties this into string theory, an aspect of physics that many believe is more mathematical play than anything connected to physical reality, which has been notably described as 'not even wrong'.
There's a lot to like, though, in this generous meander through eighteen chapters of different mathematical pondering. It'll certainly stretch your mind - and if occasional it leaves us mere mortals behind, that's not always a bad thing.
Displaying 1 of 1 review