In 1643, Descartes created one of the most important theorems of geometry. We knew he was right, but we didn’t know why. Until now

In 1643, René Descartes wrote a letter to Princess Isabel del Palatinado in which she simplified a classic problem of Western geometry and offered a solution: the so -called ‘Descartes theorem’; that, according to the famous problem that Frederick Soddy published in 1936 in Natureit can be summarized as “the sum of the squares of the four curvatures is half the square of its sum in figures.”

Basically, he found a relationship between the radios of four mutually tangent circles. The problem is that the French philosopher did not explain the reasoning behind that relationship and, in fact, he never managed to find a general formula for more than four circles. His intuition is that this solution existed, but was not able to find it. That has brought mathematicians since then.

A couple of years ago Daniel Mathews and Orion Zymaris, from the Australian University of Monash, They decided to try With a radically new approach.

What if we use tools of theoretical physics? That was the question that was asked: As Héctor Farrés explainedinstead of pulling the tools of conventional geometry, they began to play with ‘thorn’ (a type of objects of theoretical physics that need a 720 degree turn to return to their natural position).

“We use a version of thorn developed by Roger Penrose and Wolfgang Rindler, which applied to the theory of relativity,” The authors said. In this way they achieved ‘re-conceptualize’ circles as algebraic entities that can suffer from geometric transformations.

That was the key to obtaining a general formula to be able to describe increasingly complex groups of mutually tangent circles.

Why is it interesting? To start because it solves a historical problem of geometry. But, above all, because it does it again and with many ramifications.

When Andrew Wiles He managed to demonstrate Fermat’s last theoremthere was some disappointment for the use of modern mathematical tools. In that case it was understandable: part of the grace of the problem was to find the demonstration that Fermat himself said he had discovered (but never wrote).

With Descartes’s theorem is different. There was nothing to look for, just a solution to develop. And doing so shows all the potential of mathematics to destroy the limitations that lead us to grip for centuries.

In the end, As Arthur C. Clarke said“When a distinguished but elderly scientist says that something is possible, it is almost certain that he is right. When he states that something is impossible, it is almost certain that he is wrong.

Image | Frans Hals | Jacob Rus

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