04/08/2021

In Folha, Marcelo Viana discusses the unity of mathematics.

Foto: Pixabay

Reproduction of Marcelo Viana's column in Folha de S.Paulo

The Gauss-Wantzel theorem states that a regular polygon with N sides can be constructed using a ruler and compass if and only if N is the product of a power of 2 by distinct Fermat prime numbers.

Carl Friedrich Gauss (1777-1855) showed in 1798 that the construction is possible when N is of this form. Pierre Laurent Wantzel (1814-1848) confirmed in 1837 that it is impossible otherwise, as Gauss had stated without proving.

If we stop to think about it, this is an extremely surprising theorem…

Constructions using ruler and compass are at the heart of geometry, the science of forms , as conceived by classical Greece.

Prime numbers, on the other hand, are the princes of arithmetic, the science of integers, whose historical roots go back to the great civilizations of Mesopotamia and beyond.

The discovery that every integer can be written uniquely as a product of prime numbers (the fundamental theorem of arithmetic) is one of the great cornerstones of mathematics.

How is it possible that the solution to a polygon construction problem is dictated by number factorization issues? What does one thing have to do with the other?

Mathematics, so often simplistically described as "the science of numbers," encompasses geometry, arithmetic, and many other areas of knowledge: algebra, analysis, topology, probability, etc.

But, and this is perhaps its greatest fascination, mathematics also contains the study of the surprising and mysterious connections between these seemingly disparate themes, of which the Gauss-Wantzel theorem is a fine example.

That is why there are so many fields with twin names: analytical geometry, created by the French mathematician and philosopher René Descartes (1596-1650); geometric analysis, which is much more recent; algebraic topology; algebraic geometry; arithmetic geometry; and many others.

So many that a conference comes to mind, a few years ago, where a speaker made a point of explaining, with a touch of irony, that his area of research was geometric geometry…

To read the full text, visit the newspaper's website.