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

Carl Friedrich Gauß (1777-1855) showed in 1798 that building is possible when N is so. Pierre Laurent Wantzel (1814-1848) confirmed in 1837 that it was impossible otherwise, as Gauss had claimed, without proving.

If we think about it, this is an extremely surprising sentence …

Constructions with rulers and compasses are at the heart of geometry, the science of forms, as conceived in classical Greece.

Problems such as doubling the cube, dividing the angle into three, and squaring the circle followed generations of mathematicians for another two millennia until they were finally solved in the 19th century.

The cousins are the princes of arithmetic, the science of whole numbers, whose historical roots stretch back to the great civilizations of Mesopotamia and beyond.

The discovery that every integer is uniquely written as the product of prime numbers (the principle of arithmetic) is one of the great foundations of mathematics.

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

Mathematics, so often referred to simply as “the science of numbers”, includes geometry, arithmetic, and many other fields of knowledge: algebra, calculus, topology, probability, and so on.

But, and in this perhaps greatest fascination, mathematics also includes the study of the surprising and mysterious connections between these apparently so different subjects, of which Gauss-Wantzel’s theorem is a fine example.

This is why there are so many domains with double names: analytical geometry, created by the French mathematician and philosopher René Descartes (1596-1650); geometric analysis, much newer; algebraic topology; algebraic geometry; arithmetic geometry; and many others.

So many that a few years ago a conference comes to mind where a speaker stressed with some irony that his research area was geometric geometry …

The best part is that the discovery of such compounds remains a fertile area of research, with applications in physics, for example, today.

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