Physicists Crack 380-Year-Old Descartes Circle Problem

The field of geometry has just experienced a significant advancement.

Mathematicians from Monash University have solved an ancient riddle that dates back over hundreds of years to the 1600s, expanding Descartes' Circle Theorem into uncharted territories.

Leveraging sophisticated mathematical techniques borrowed from physics, the researchers have formulated a comprehensive equation applicable to multiple tangent circles, providing new perspectives on an equation initially suggested by mathematician René Descartes.

Descartes' theorem, a fundamental principle in geometry, elucidates the connection between four circles that are all touching each other.

However, for centuries, extending the equation to encompass more than four circles has baffled mathematicians—until now.

Associate Professor Daniel Mathews from Monash University’s School of Mathematics, together with PhD candidate Orion Zymaris, have discovered the mathematical formula behind "n-flowers." These intricate geometrical designs emerge when numerous circles touch each other in specific arrangements.

In the realm of circle packing theory, flowers act as a basic structural element.

It is widely recognized that once the curvatures of the outer circles (petals) within an n-flower are known, the curvature of the inner circle can be accurately calculated.

The research was grounded in contemporary mathematical methods that involve spinors—mathematical constructs found not only in quantum mechanics but also in relativity theory.

From Descartes to spinors

In 1643, Descartes presented a challenge to Princess Elisabeth of the Palatinate, believing he could conquer it easily. Having recently developed Cartesian coordinates, he thought nothing would be too difficult. However, he stumbled upon finding a solution at first. It was only after refining the problem into something more pragmatically approachable that his work evolved into what we now recognize as the classic Descartes Circle Theorem," explained Mathews. release .

The equation was initially introduced by Yamaji Nushizumi in 1751 and has since been independently rediscovered several times—at different points, Jacob Steiner encountered it again in 1826, William Beecroft stumbled upon it in 1842, and Frederick Soddy revisited it with his poetic interpretation in 1936.

Other researchers have generalized the outcome in various manners; however, this marks the initial expansion providing a precise equation linking the radii of any number of circles within a two-dimensional space.

Zymaris, who conducted the groundbreaking research for her doctoral thesis, emphasized the surprising links to various areas of mathematics and physics.

"Our method employed sophisticated geometric techniques influenced by physics, which turned out to be quite unexpected," he stated.

He mentioned that spinors, which are frequently utilized in quantum mechanics, were central to their method.

We utilized a variant of spinors crafted by Nobel laureate Roger Penrose and Wolfgang Rindler, applying them to the theory of relativity.

It turns out that the mathematical frameworks used for describing quantum spin and relativity also assist in understanding circle packings.

A query over 380 years old finally gets answers

The project not only represents a substantial advancement in the realm of pure science. 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 finding exemplifies how classic issues can lead to the development of new mathematical concepts even centuries afterward," stated Mathews.

It's astounding to consider that a query which troubled Descartes in the 1600s continues to hold fresh solutions yet to be uncovered.

The groundbreaking study has been released in the Journal of Geometry and Physics .