380-year-old Descartes circle problem finally solved with help from physics
The world of geometry just witnessed a pivotal breakthrough.
Mathematicians at Monash University have cracked a centuries-old puzzle dating back to the 17th century, extending Descartes’ Circle Theorem into a bold new territory.
Using advanced mathematical tools inspired by physics, the team has derived a general equation for any number of tangent circles, offering fresh insights into an equation originally proposed by mathematician René Descartes.
Descartes’ theorem, a cornerstone of geometry, defines the relationship between four mutually tangent circles.
But for centuries, generalizing the equation to more than four circles had eluded mathematicians—until now.
Associate Professor Daniel Mathews from Monash University’s School of Mathematics, along with PhD candidate Orion Zymaris, has identified the equation that governs “n-flowers”—the complex geometric patterns formed by larger configurations of tangent circles.
In circle packing theory, flowers serve as a fundamental building block.
It is well established that once the curvatures of the outer circles (petals) in an n-flower are known, the curvature of the central circle can be precisely determined.
The researchers based their study on modern mathematical techniques involving spinors—mathematical entities that also appear in quantum mechanics and relativity.
From Descartes to spinors
“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,” Mathews said in a release.
The equation was first proposed by Yamaji Nushizumi in 1751 and has been independently rediscovered multiple times—by Jakob Steiner in 1826, William Beecroft in 1842, and Frederick Soddy in 1936, who famously rephrased it in the form of a poem.
“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.
He explained that spinors, commonly used in quantum mechanics, played a key role in their approach.
“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 380-year-old question finds answers
The work not only marks a significant step forward in pure mathematics but also showcases the growing strength of the topology group at Monash University, which now includes nine PhD students—five of them women.
“This discovery is an exciting example of how classical problems can inspire new mathematics centuries later,” Mathews said.
“It’s incredible to think that a question Descartes struggled with in the 1600s still has new answers waiting to be found.”
Source: Interesting Engineering
Scientists Just Discovered Quantum Signals Inside Life Itself
380-year-old Descartes circle problem finally solved with help from physics