In Shape, Jordan Ellenberg codifies geometry, explores its place in human imagination

“Girls can’t do Euclid: can they, sir?” The Mill on the Floss contains one of George Eliot’s sharpest caricatures in the figure of the foul schoolmaster Stelling. About girls, he reassures his young charges: “They’ve a great deal of superficial cleverness; but they couldn’t go far into anything.” Certainly not geometry, that maker of men. Stelling embodied British pedagogy at the time, with all its complacent sexism and emphasis on rote memorisation. But as the emphasis shifted from students parroting proofs to forming their own, geometry remained exalted for its power to cultivate deductive reasoning, to toughen and refine the mind. “I keep waiting for that to happen to me and it never has,” mathematician Jordan Ellenberg confesses in his unreasonably entertaining new book, Shape, with its modest subtitle: “The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else.” In granular detail, he reveals how geometric thinking can allow for everything from fairer US elections to better pandemic planning. Before we begin: A moment of appreciation for the popular math writer who must operate with the same stealth, balletic improvisation and indomitable self-belief as someone trying to corner a particularly skittish and paranoid cat into the pet carrier. No sudden moves! Approach carefully; compliment liberally — precious reader, brilliant reader. Offer bribe and blandishment. Assure us it won’t hurt. Ellenberg, a professor at the University of Wisconsin–Madison, is rather spectacular at this sort of thing. A seam in his narrative is a critique of how math, and especially geometry, has been taught. (His strategy for success in teaching is to employ more strategies; multiply approaches so students might find one that works for them.) He also takes a few well-aimed swipes at current depictions of the campus culture wars. The “cosseted” American college student might have launched a thousand Substacks, but have you heard of the “Conic Sections Rebellion”? Some 44 students, including the son of Vice President John Calhoun, were expelled from Yale in 1830 for refusing to take a geometry exam. Geometry occupies a peculiar place in the imagination. “There are people who hate it,” Ellenberg writes, “who tell me geometry was the moment math stopped making sense to them. Others tell me it was the only part of math that made sense to them. Geometry is the cilantro of math. Few are neutral.” And yet we are wired for it: “From the second we exit hollering from the womb we’re reckoning where things are and what they look like.” You can give babies geometry tests. If you offer them pictures of pairs of shapes, most of them identical, but occasionally with one of the shapes reversed, babies will stare longer at the reversed shapes: “They know something’s going on, and their novelty-seeking minds strain toward it.” (Full disclosure: I was not able to replicate this finding. My subject proved recalcitrant and ate said card, offering a twist, perhaps, on Ellenberg’s notion that geometry is “primal, built into our bodies.”) Those who drink the hallucinogenic ayahuasca report seeing two-dimensional patterns or throbbing, three-dimensional hexahedral cells. When the reasoning mind melts away, only shapes remain. Geometry gives us a world unclad. “Euclid alone has looked on Beauty bare,” wrote Edna St Millay. That feeling of mystical revelation — of a shimmering, underlying order that we can apprehend if we purify our perception — might explain the mutual affinity between poets and geometers. Dante mentions squaring the circle in Paradiso. Wordsworth repeatedly invokes Euclid. Many of the mathematicians cited in Ellenberg’s book wrote verse. Ellenberg’s preference for deploying all possible teaching strategies gives Shape its hectic appeal; it’s stuffed with history, games, arguments, exercises. One entire lesson hinges on the question: How many holes are there in a pair of pants — one, two or three? Ellenberg puts footnotes to their only acceptable, nonacademic use, which is jokes. If your grasp on the Virahanka-Fibonacci sequence is as hazy as mine, the biographical sections are honey. What a parade of beautiful minds, splendid eccentrics, catty squabbles. We meet the “mosquito man,” Sir Ronald Ross, whose study The Logical Basis of the Sanitary Policy of Mosquito Reduction became the foundation of the so-called random walk theory. And the polymathic Johann Benedict Listing, one of those miraculous dabblers that the 19th century seemed to churn out, who flitted from measuring the Earth’s magnetic field to sugar levels in the urine of diabetic patients. Above all, Ellenberg borrows from one of the greatest math teachers — I refer, of course, to Mrs Whatsit from A Wrinkle in Time — and embeds his approach in a narrative, not of the history of geometry but of our old association with it, of mathematics as a kind of mother tongue. You might balk at delving into eigenvalues — “that strangely complicated number that governs the rate of geometric growth” — but I’ll bet you can recognise the sunny confidence of a C major chord and its individual notes. “The geometry was there in our bodies,” Ellenberg writes, “before we knew how to codify it on the page.” For all Ellenberg’s wit and play (and his rightful admiration of some excellent 19th-century beards), the real work of Shape is in codifying that geometry on the page. Ellenberg butters you up to put you to work. I applied myself to my scrap paper with all the passionate ineptitude I remembered from my school days. The math he presents is serious and demanding and — this is key — shaping the world around us, from our understanding of the spread of COVID-19 to gerrymandering. Wordsworth imagined that Euclidean geometry “wedded soul to soul in purest bond / Of reason, undisturbed by space or time.” To Ellenberg, geometry is not a reprieve from life but a force in it — and one that can be used for good, ill and for pleasures of its own. It binds and expands our notions of the world, the web of the real and the abstract. “I prove a theorem,” poet Rita Dove wrote, “and the house expands.” Parul Sehgal c.2021 The New York Times Company