‘Shape’ Makes Geometry Entertaining. Really, It Does.

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 recognize 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.

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