The Gauss-Wantzel theorem states that a regular polygon with N sides can be constructed using a straightedge and compass if and only if N is the product of a power of 2 by distinct Fermat primes.
Carl Friedrich Gauss (1777-1855) showed in 1798 that construction is possible when N is that way. Pierre Laurent Wantzel (1814-1848) confirmed in 1837 that it is otherwise impossible, as Gauss had claimed without proving.
If we stop to think, this is an extremely surprising theorem…
Constructions with a ruler and compass are at the heart of geometry, the science of forms, as conceived by classical Greece.
Problems like doubling the cube, trisecting the angle, and squaring the circle haunted generations of mathematicians for another two millennia, until they were finally resolved in the 19th century.
The cousins are the princes of arithmetic, the science of whole numbers, whose historical roots go back to the great civilizations of Mesopotamia and beyond.
The discovery that every whole number is uniquely written as a product of prime numbers (the fundamental theorem 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 questions of number factorization? 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, and so on.
But, and in this perhaps its greatest fascination, mathematics also contains the study of the surprising and mysterious connections between these apparently so disparate subjects, of which the Gauss-Wantzel 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 René 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 remains a fruitful field of research, with applications, for example, in physics today.
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