Mathematicians find 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 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."

"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. 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."

"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."

For over two thousand years, mathematicians have developed thousands of equations to calculate pi's digits more efficiently. These formulas appeared as isolated discoveries with no apparent connections. Recent research has revealed that centuries of pi formulas are part of a single, previously hidden mathematical structure. Archimedes pioneered pi calculation using geometric polygons, establishing bounds between 3.140845 and 3.142857. Around the 14th century, Madhava of Sangamagrama introduced the first exact formula using infinite series, though it converged slowly. Leonhard Euler later contributed additional discoveries. This unified framework now connects all these historically separate mathematical achievements.

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