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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 now 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.
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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.
Then, around the 14th century, Indian mathematician Madhava of Sangamagrama provided the first exact formula, expressed as an infinite series—a sum of endlessly many terms that, if you could somehow add them all up, would yield pi exactly. The catch: his series converged agonizingly 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 1900s, the mathematician Srinivasa Ramanujan produced formulas that are still revered for their efficiency today.
Amanda Montañez; Source: “From Euler to AI: Unifying Formulas for Mathematical Constants,” by Tomer Raz et al. Preprint posted November 16, 2025 to 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 that you get to cite 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 head Ido Kaminer’s 2019 Ramanujan Machine, an AI bot that seeks out new conjectures for calculating mathematical constants. Anyone can download the software for free, and many have used it to find new pi formulas to join the heap. The bot’s unconventional approach was a viral success, if not taken entirely seriously by mathematicians. “When we started doing AI research in this area of math,” Kaminer says, “it was seen as a fringe idea.”
But as the machine and other mathematicians kept churning out formulas, eventually the question became unavoidable: Were any of them connected?
The group, who also have backgrounds in areas such as physics and math, approached the problem like experimentalists and decided to gather a dataset. Tomer Raz, then a master’s student at Technion, wrote code to download every math paper that had ever been uploaded to the preprint server arXiv.org, running his laptop seven days a week, 24 hours a day, for six weeks to download 455,050 papers at a slow enough rate to respect the website’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 the CMF, which some members of Kaminer’s group had introduced in 2023. Shalyt calls it a Swiss army knife for mathematics. “It can unify 2,000-year-old formulas [and] give hierarchy for constants in math, and we hope to [use it to] prove some properties of irrationality related to the Riemann hypothesis,” he says.
Think of the CMF like 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, the 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), however mismatched they appear on the surface.
The group derived the CMF of pi, then used algorithms to see where each formula fit inside 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 descend from a single CMF. Another 51 percent belong to broader clusters. (The researchers are still working out their precise relationships.) Only 6 percent of the formulas remain orphans, with no proven connection to anything else.
It’s an open question whether a more complex CMF could capture the entire set, Kaminer says. Another open question is whether every single equation generated from the 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 wasn’t involved in the study (though 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 been discovering atomic elements one by one “and then all of a sudden, someone let loose a computer program that constructed the whole periodic table automatically.”
Mathematician George Andrews, a professor emeritus at the Pennsylvania State University (who famously uncovered a lost trove 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 mathematics done in a serious way,” he says. “More and more surprising things should emerge.”