The Gauss-Wantzel theorem states that a regular polygon with N sides can be constructed using a straight interval and compass if and only if N is the product of a power of 2 by the characteristic Fermat prime.
Carl Friedrich Gauss (1777-1855) showed in 1798 that construction is possible when N is this way. Pierre Laurent Wentzel (1814-1848) asserted in 1837 that it was impossible otherwise, as Gauss claimed without proof.
If we stop to think, this is a very surprising theory…
Ruler and compass constructions are at the heart of geometry, the science of forms, as envisioned by classical Greece.
Problems such as doubling the cube, triple-angle, and squaring the circle haunted generations of mathematicians for another two millennia, until they were finally solved in the 19th century.
Cousins are lords of arithmetic, the science of integers, whose historical roots go back to the great civilizations of Mesopotamia and beyond.
The discovery that every integer is uniquely written as a product of prime numbers (the fundamental theorem of arithmetic) is one of the great foundations of mathematics.
How can the solution to the polygon construction problem be dictated by numerical analysis questions? What does one thing have to do with the other?
Mathematics, often described in simplified form as “numerology”, contains geometry, arithmetic, and many other areas of knowledge: algebra, analysis, topology, probability, and so on.
But, and in this perhaps its greatest charm, mathematics also contains the study of the surprising and mysterious connections between these seemingly disparate subjects, of which Gauss and Inzel’s theorem is a good example.
That is why there are many fields with dual names: analytic geometry, created by the French mathematician and philosopher René Descartes (1596-1650); engineering analysis, much newer; Algebraic topology, algebraic geometry, arithmetic geometry, and many other things.
So much that a conference came to mind a few years ago, where the speaker explained, with some irony, that his field of research was engineering engineering…
Best of all, the discovery of such connections remains a fruitful area of research, with applications, for example, in physics today.
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