Descartes’ Unfinished Mystery: Mathematicians Solve 380-Year-Old Geometry Problem

Monash University researchers have extended Descartes’ Circle Theorem by finding a general equation for any number of tangent circles, using advanced mathematical tools inspired by physics.

A centuries-old geometric puzzle dating back to the 17th century has finally been solved by mathematicians at Monash University, offering new insights into an equation originally proposed by philosopher and mathematician René Descartes.

Their breakthrough, published in the Journal of Geometry and Physics, expands on the well-known Descartes Circle Theorem, which defines the relationship between four mutually tangent circles. While the theorem has long been a cornerstone of geometry, mathematicians had struggled for generations to generalize it to configurations involving more than four circles—until now.

Associate Professor Daniel Mathews, a mathematician at Monash University School of Mathematics, and PhD candidate Orion Zymaris have found the equation that governs these larger patterns of tangent circles, known as “n-flowers.”

Their proof, which draws on modern mathematical techniques involving spinors—objects that also play a role in quantum mechanics and relativity—solves a problem that has remained open for more than 380 years.

Revisiting Descartes’ Challenge

“Descartes posed a problem to Princess Elisabeth of the Palatinate in 1643, assuming he could solve it. After all, he had just invented Cartesian coordinates! But he couldn’t, and when he revised the problem to a practically solvable one, this has become known as the classic Descartes Circle Theorem,” said Associate Professor Mathews. “Others have generalised the result in other ways, but this is the first extension of the result to give an explicit equation relating the radii of an arbitrary number of circles in the plane.”

Zymaris, whose PhD research led to the breakthrough, highlighted the unexpected connections to other fields of mathematics and physics.

“Our approach used advanced geometric tools inspired by physics, which was surprising,” he said. “Spinors are widely used in physics, especially in quantum mechanics. We used a version of spinors developed by Nobel prize-winner Roger Penrose, and Wolfgang Rindler, which they applied to the theory of relativity.”

“It turns out that the same mathematical structures that describe quantum spin and relativity also help us understand circle packings.”

A Victory for Pure Mathematics and a Growing Team

The work not only advances pure mathematics but also highlights the growing strength of the topology group at Monash University, which now includes nine PhD students—five of whom are women.

“This discovery is an exciting example of how classical problems can inspire new mathematics centuries later,” said Associate Professor Mathews. “It’s incredible to think that a question Descartes struggled with in the 1600s still has new answers waiting to be found.”

Reference: “Spinors and the Descartes circle theorem” by Daniel V. Mathews and Orion Zymaris, 25 February 2025, Journal of Geometry and Physics.
DOI: 10.1016/j.geomphys.2025.105458

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