For more than two millennia, mathematicians have produced a growing pile of pi equations in their ongoing search for methods to calculate pi faster and faster. The stack of equations has now grown to thousands, and algorithms can now generate an infinity. Each discovery has come alone, as a fragment, without any obvious connection to the others. But now, for the first time, centuries of pi formulas have been shown to be part of a unified, previously hidden structure.
Divide the circumference of a circle by its diameter and you get pi. But what are the numbers really? Measuring physical circles won’t tell you – your tools are too clumsy to detect piss endless numbers. 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 of pi’s value. He thought of a circle as an infinite polygon with sides of zero length. The mathematics to handle infinitesimals (calculus) would not 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…, capturing it in a range. His austerity lasted for 1600 years.
On supporting science journalism
If you like this article, please consider supporting our award-winning journalism by subscribes. By purchasing a subscription, you help secure the future of impactful stories about the discoveries and ideas that shape our world today.
Then, around the 14th century, the Indian mathematician Madhava of the Sangamagrama gave the first exact formula, expressed as an infinite series—a sum of infinitely many terms that, if you could somehow add them all together, would give pi exactly. The catch: his series converged painfully slowly, requiring hundreds of terms just to nail down a few decimal places. More than three hundred years later, Leonhard Euler discovered another series that converged faster. And in the early 20th century, the mathematician Srinivasa Ramanujan produced formulas that are still revered for their effectiveness today.
Amanda Montañez; Source: “From Euler to AI: Unifying Formulas for Mathematical Constants,” by Tomer Raz et al. Preprint posted 16 November 2025 at https://arxiv.org/pdf/2502.17533 (reference)
Each equation seemed unrelated to the others. But in late 2025, 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. “It’s not every day you can quote Archimedes,” says Ph.D. student Michael Shalyt, part of the team. The structure, called a conservative matrix field, or CMF, acts as a kind of mathematical common ancestor, showing how formulas that look nothing alike turn out to be different expressions of the same underlying object.
The project grew out of group leader Ido Kaminer’s 2019 Ramanujan Machine, an AI bot that searches for new conjectures to calculate mathematical constants. Anyone can download the software for free, and many have used it to find new pi formulas to join the pile. Boten’s unconventional approach was a viral success, if not taken entirely seriously by mathematicians. “When we started doing AI research in this area of mathematics,” says Kaminer, “it was seen as a fringe.”
But as the machine and other mathematicians continued to figure out formulas, the question eventually became inevitable: Were any of them connected?
The group, which also has a background in fields such as physics and mathematics, approached the problem as experimental and decided to collect a data set. Tomer Raz, then a master’s student at the Technion, wrote code to download every math paper that had ever been uploaded to the arXiv.org preprint server, and ran his laptop 24/7 for six weeks to download 455,050 papers at a speed slow enough to respect the site’s limit.
The group then deployed GPT-4o in combination with specialized algorithms to detect pi-related equations, translate them into executable code, and remove trivial duplicates. From nearly half a million papers, they extracted 385 unique formulas, including about 10 percent that originated from the Ramanujan machine.
For the next step, they recast the 385 equations into the same format – a special type of infinite series. But the expressions still all converged to pi, leaving no obvious way to compare them. Something deeper was needed.
That something was CMF, which some members of Kaminer’s group had introduced in 2023. Shalyt calls it a Swiss army knife of mathematics. “It can unify 2,000-year-old formulas (and) provide hierarchy for constants in mathematics, and we hope to (use it to) prove some properties of irrationality related to the Riemann hypothesis,” he says.
Think of CMF as gravity defined on a grid. Each pi formula traces a different path across the grid. Just as a gravitational field guarantees that the energy difference between two points is the same, regardless of route, CMF guarantees that only the destination matters. From this single constraint something remarkable emerges: when two pi formulas trace parallel paths through the same CMF grid, they are equivalent (one can be transformed into the other), no matter how incongruous they appear on the surface.
The group derived the CMF for pi, then used algorithms to see where each formula fit into the grid, finding clusters of similar equations. An algorithm formally proved whether a cluster of equations belonged to the CMF. The result: 43 percent of all known pi formulas originate from a single CMF. A further 51 per cent belong to wider clusters. (The researchers are still working out their exact ratios.) Only 6 percent of the formulas remain orphans, with no proven connection to anything else.
It is an open question whether a more complex CMF can capture the whole set, says Kaminer. Another open question is whether every single equation generated from CMF is a pi formula – so far all the equations the team has tried have worked.
David Bailey, a retired computer scientist formerly at Lawrence Berkeley National Laboratory who was not involved in the study (although a pi formula bears his name and the group used one of his algorithms), says the project’s results are as if 17th-century chemists had discovered atomic elements one by one “and then someone unleashed the whole thing on a computer.” periodic table automatically.”
Mathematician George Andrews, a professor emeritus at Pennsylvania State University (who famously unearthed a lost collection of Ramanujan’s notes) had previously criticized the group for naming their machine after Ramanujan. But he had nothing but praise for the current work. “This is serious math done in a serious way,” he says. “More and more surprising things should appear.”
