"One can’t help feeling that, in those opening years of the 1900s, something was in the air," writes mathematician Jordan Ellenberg. It’s page 90 of Shape: The Hidden Geometry of Absolutely Everything, and he’s launching into the second act of his complex history of geometry (think "History of the World in 100 Shapes"). For page after page, we’ve been introduced to symmetry, to topology, and to the kinds of notation that make sense of complex questions like "how many holes has a straw"? Now, though, the gloves are off, as Ellenberg records the fin de siecle’s "painful recognition of some unavoidable bubbling randomness at the very bottom of things".

Normally when sentiments of this sort are trotted out, they’re there to introduce readers to the wild world of quantum mechanics. Quantum has such a grip on our imagination that we tend to forget that, when Niels Bohr, Werner Heisenberg and others developed it in the 1920s, it was the johnny-come-lately icing on an already fairly indigestible cake.

A good 20 years before physical reality was shown to be unreliable at small scales, mathematicians were pretzeling our very ideas of space. They had no choice: at the Louisiana Purchase Exposition in 1904, Henri Poincare, by then the world’s most famous geometer, described how he was trying to keep reality stuck together in light of James Clerk Maxwell’s famous equations of electromagnetism. In that talk, he came startlingly close to gazumping Einstein to a theory of relativity.

Also at the same exposition was Sir Ronald Ross, who had discovered that malaria was carried by the bite of the anopheles mosquito. He baffled many with his presentation of a mathematical model of disease transmission – the one we use today to predict just about everything, from pandemics to political elections.

It’s hard to imagine two mathematical talks less alike than those of Poincare and Ross. And yet they had something vital in common: both shook their audiences out of mere three-dimensional thinking. And thank goodness for it: Ellenberg takes time to explain just how restrictive Euclidean thinking is.

For Euclid, the first geometer, living in the 4th century BC, everything was geometry. When he multiplied two numbers, he thought of the result as the area of a rectangle. When he multiplied three numbers, he called the result a "solid". Euclid’s geometric imagination gave us number theory; but tying mathematical values to physical experience locked him out of more or less everything else. Multiplying four numbers? Now how are you supposed to imagine that in three-dimensional space?

For the longest time, geometry seemed exhausted: a mental gym; sometimes a branch of rhetoric.

But the more dimensions you add, the more capable and surprising geometry becomes. And this, thanks to runaway advances in our calculating ability, is why geometry has become our go-to manner of explanation for, well, everything. For games, for example: and extrapolating from games, for the sorts of algorithmical processes we saddle with that profoundly unhelpful label "artificial intelligence".

All game-playing machines share the same ghost, the "Markov chain", formulated in 1913 by Andrey Markov and used by him to map the probabilistic landscape generated by sequences of likely choices. Markov used his eponymous chain, rhetorically, to strangle religiose notions of free will in their cradle.

From isosceles triangles to free will is quite a leap, and by now you will surely have gathered that Shape is anything but a straight story. That’s the thing about maths: it does not advance; it proliferates.

Containing multitudes as he must, Ellenberg’s eyes grow wider and wider, his prose more and more energetic, as he moves from what geometry means to what geometry does in the modern world.

I mean no complaint when I say that, by two thirds of the way in, Ellenberg comes to resemble his friend John Horton Conway, the game-playing, toy-building celebrity of the maths world, who died from Covid-19 last year.

"He wasn’t being wilfully difficult; it was just the way his mind worked...," writes Ellenberg. "You asked him something and he told you what your question reminded him of."

This is why Ellenberg takes the trouble to draw a mind map at the start of Shape. This, and the index, offer the interested reader a whole new way ("more associative than deductive") of rereading the book.

Writing for a non-mathematical audience, Ellenberg is a popular educator at the top of his game.

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