One Pi Formula to Rule Them All

For more than two millennia, mathematicians have produced a growing heap of pi equations in their ongoing search for methods to calculate pi faster and faster. The pile of equations has grown into the thousands, and algorithms now can generate an infinitude. Each discovery has arrived alone, as a fragment, with no obvious connection to the others. But now, for the first time, centuries of pi formulas have been shown to be part of a unified, formerly hidden structure.

Divide any circle’s circumference by its diameter and you get pi. But what, exactly, are its digits? Measuring physical circles won’t tell you—your tools are too clunky to discover pi’s endless numerals. Uncovering its true value requires something much more powerful: a formula.

It all started with Archimedes, who developed the world’s first known mathematical proof for pi’s value. He thought of a circle as an infinite-sided polygon with sides of zero length. The math to handle infinitesimals (calculus) wouldn’t arrive for another 1,900 years, so instead he circumscribed 96-sided polygons on the outside and inside of a circle and used geometry to calculate their perimeters. He was able to determine that pi fell somewhere between 3.140845… and 3.142857…, trapping it in a range. His rigor stood for 1,600 years.

A team of seven AI researchers at the Technion – Israel Institute of Technology found a previously unknown mathematical structure underlying hundreds of pi formulas, including those of Archimedes, Euler and Ramanujan.