Pythagoras's Trousers

Me, "We", and my secret math entanglement with Russia

“The world is bound by secret knots” – Athanasius Kircher (1602-1680)

Last week, through the magic portal that is the Internet, I received an email from a scholar in the “Department of General and Comparative Historical Linguistics” at Lomonosov Moscow State University. The writer, Daria Studenikina, suspected I had knowledge of colloquial Russian mathematics terminology. She was seeking clarification. 30 years ago I wrote a book titled Pythagoras’s Trousers, a term, Ms. Studenikina now informed me, is the nick-name Russians use for the Pythagorean Theorem. Because, as she put it, the theorem “is proved by using the equality of the sum of the squares built on the legs of a right triangle, and the area of a square built on the hypotenuse of that triangle (the drawing looks like pants).”

How did I, as an English speaker, know of this sartorial Russian metaphor?

I did not. I’d dreamed up my title in ignorance of any Slavic connection as an amusing summation of my book – a history of physics focusing on the role Pythagorean thinking has played in the 2,500 year long tradition of this science. Pythagoras gave the Western world the idea that reality could be described by mathematics, what he famously termed “the music of the spheres.” Though other cultures have developed math, no other society we know of has seen in it the foundation for a philosophy of nature. Galileo, Newton, Einstein, and Stephen Hawking were all searching for some version of Pythagoras’s “cosmic harmonies.”

But why did I add “‘trousers” to my title?

Because the book is also an exploration of how physics has been an overwhelmingly male practice from which women have been excluded on both philosophical and practical levels. In fifth century BC Greece, the Pythagoreans viewed math as a male activity – and so it has largely been construed ever since. I considered calling my book Pythagoras’s Pants, but I liked the ring of “Trousers” better, with its hint of the ridiculous and a whiff of pomposity. My editor joked that one day I’d find it in the fashion section of a bookstore. And to my delight I did.

Years after the book was published, I was astounded to learn of a secret connection between my title and the Russian writer Zamyatin’s remarkable sci-fi classic We, a dystopian masterpiece believed to have been an inspiration for both Huxley’s Brave New World and Orwell’s Nineteen Eighty-Four. Written in 1920 as a scathing satire of the Soviet State, We is set in a future society controlled by logic and ‘reason.’ Individual behavior is dictated by algorithms; people are known by numbers rather than names; even music is composed along mathematical lines. At one point, Zamyatin’s hero, D-503 (a spacecraft engineer), attends a state sponsored symphony – its title: “Pythagoras’s Trousers.”

I stumbled upon this extraordinary coincidence after hearing a piece of music with the same name by the avant-pop British band Penguin Cafe Orchestra. Its founder, Simon Jeffes, had composed his piece in homage to Zamyatin, to bring the latter’s fictional work into the realm of the real.

In my response to Ms. Studenikina I mentioned the We connection. Was she aware of it? Most certainly she was. In her research, Studenikinia told me, she is “analyzing Zemyatin’s translations into English” and has been intrigued by the fact that the first translation of We uses the word “trousers” in relation to the symphony when, in her professional opinion, the more literal term would be “pants.” That translation, she opined, “was made by a person far from translation theory.”

Thus the bond between me and the Soviet master results from a case of linguistic amateurism. “He did his best,” Studenikina adds.

So now after years of wonderment, I at last understand the naming of Zamyatin’s musical offering: it wasn’t mere fancy.

The story get curiouser, for there is yet another mathematical entanglement afoot. While the Pythagorean Theorem may be the most famous rule in math, it is not universally true. It’s only true in Euclidean geometry; not in either spherical or hyperbolic geometry. And as it happens I’ve spent an inordinate amount of time pondering the parameters of hyperbolic space. In my Crochet Coral Reef project – a science+art endeavor now encompassing nearly 30,000 contributors across 52 cities worldwide – we craft corals inspired by hyperbolic geometry.

Crocheted models of hyperbolic surfaces by the author.

In planar, or Euclidean, geometry the angles of a triangle always add up to 180˚ and this is required for Pythagoras’s theorem to hold . On the two other possible 2D geometries – the surface of a sphere and the so-called hyperbolic plane – angles in a triangle don’t sum to 180˚. On a sphere they add up to more, on a hyperbolic plane to less. Thus on both surfaces, math’s most seemingly ingrained rule doesn’t apply. Zamaytin’s dissident protagonist, with his determination to break free from the rule-bound-restrictions of his hyper-ordered life, would surely have approved. Indeed mathematicians who discovered this geometry were nearly driven mad by its aberrations. “Fear it no less than the sensual passions, for it will deprive you of your health, peace of mind, and happiness in life,” wrote one early pioneer of hyperbolic math to another.

Sea creatures such as corals and kelps and nudibranchs have been making hyperbolic forms in the fibers of their beings since the Silurian age, but for humans it’s not so easy. It turns out, we can crochet hyperbolic surfaces to create soft, floofy models of a structure that mathematicians long thought impossible. (Here’s a nice explainer video). Then you can stitch theorems onto these surfaces to demonstrate how basic geometric laws are different in hyperbolic space: how the angles of a triangle can add up to zero degrees, and how parallel lines can diverge.

You can also roll up such a surface to create the hyperbolic equivalent of a tube, which has a single opening at the top and two openings below – so it look like, well… a pair of pants. Indeed, mathematicians call such forms hyperbolic pants.

Dr. Daina Taimina, who discovered hyperbolic crochet, has made a version of this object which you can see below.

Non-Pythagorean hyperbolic pants (standing upright), by Dr. Daina Taimina.