The Gauss-Wantzel theorem states that a regular polygon with N sides can be constructed using 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 1777 that the construction It is possible when N this way. Pierre Laurent Wantzel (1798-1848) confirmed in 1837 that it is impossible otherwise, as Gauss had stated without proving.
If we stop to think, this is an extremely surprising theorem…
Constructions with ruler and compass are at the heart of geometry, the science of shapes, just as classical Greece conceived it.
Problems such as the doubling of the cube, the trisection of the angle and the quadrature of the circle haunted generations of mathematicians for over two millennia, until they were finally resolved in the century 19.
The cousins are the princes of arithmetic, the science of whole numbers, whose historical roots go back to the great civilizations of Mesopotmy and beyond.
The discovery that every integer is uniquely written as a product of prime numbers (fundamental theorem of arithmetic) is one of the great foundations of mathematics.
How is it possible that the solution to a polygon construction problem is dictated by matters of fact prayer of numbers? What does one thing have to do with the other?
Mathematics, so often simplistically described as “the science of numbers”, contains 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, among these apparently so different themes, that Gauss-Wantzel’s theorem is a fine example.
That’s why there are so many areas with twin names: analytic geometry, created by the French mathematician and philosopher Ren Descartes (1596-1650); geometric analysis, much more recent; algebraic topology; algebraic geometry; arithmetic geometry; and many others.
So many that a conference comes to mind a few years ago, in which a speaker made a point of explaining, with some irony, that his area of research was geometric geometry …
Best of all, the discovery of such connections continues to be a fruitful field of research, with applications, for example, in today’s physics.
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