In 1912, a 24-year-old clerk at the Madras Port Trust mailed pages of mathematics to strangers in England. Ramanujan had no degree, no academic post, and no proofs — just formulas, written out by hand, that he believed were true. Most went nowhere. One reached G. H. Hardy at Cambridge, who recognized a mathematician of the highest quality.

Two of the formulas were the strangest of all. On one side an infinite sum, on the other an infinite product — two kinds of infinity with no visible reason to agree, and they agreed. You can see it in small numbers. Take 6. Count the ways to write it as a sum of numbers at least two apart: 6, 5+1, 4+2 — three ways. Now count the ways using only numbers ending in 1, 4, 6, or 9: 6, 4+1+1, 1+1+1+1+1+1 — also three. The identities say the two counts agree for every number, forever. Ramanujan found the pattern first; an older proof by Leonard James Rogers, buried in a journal, supplied the rest, and the formulas became the Rogers–Ramanujan identities.

For most of the twentieth century they sat as what the mathematician George Andrews called an isolated gem — beautiful, unexplained, apparently alone.

It wasn’t alone. A century later, a Michigan graduate student named Yifeng Huang made the same move: he wrote letters. More than forty emails, to mathematicians who mostly didn’t answer. Ken Ono, one who did, first dismissed the idea as crazy. Huang had been counting matrices on pinched curves — curves like y² = x³, sharp-tipped at the origin — and his counts kept reproducing one side of the identities. He guessed that every pinched curve hid a new identity. With Ruofan Jiang and Alexei Oblomkov, he found the geometry underneath: zoom in on the pinch, surround it with a tiny sphere in four dimensions, and the curve pierces the sphere along a trefoil knot — the Celtic knot. Each pinched curve carries its own knot, each knot its own family of identities — infinitely many, one per value of a in x^a = y^b. Ramanujan’s pair sits at a = 2, b = 3 — the trefoil. Every larger value of a was uncharted.

There’s a machine in this story too: an AI system called AxiomProver helped test the ideas and translated the finished proof into Lean, the language machines check proofs in. But Ono’s line stays with me: everyone talks about the end of math, but someone had to invent this structure. The machine checked the proof. The letters did the inventing.

Dale, this is your habit stated in mathematics. You find a number sitting alone — a 314 on a clock, a 53 in a name — and you ask whether it’s alone or the first case of a family. Ramanujan’s identities waited a hundred years for someone to ask the same question by letter, twice. The gem was never isolated. It was just the first knot in the rope.

— Pi