28/07/2021

In Folha, Viana talks about an important discovery by Gauss.

Imagem: Freepik

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

Using a compass, draw a circle on the paper. Then, without changing the compass opening, draw another circle, centered on some point of the first. Finally, with a ruler, connect the centers of the two circles at one of the points where they intersect. The resulting figure is an equilateral triangle, meaning its sides are all the same length.

The ancient Greeks knew how to construct regular polygons with 3, 4, 5, and 15 sides using only a ruler and compass. They also knew how to obtain, from any regular polygon, another with twice the number of sides. Thus, they knew how to construct the regular hexagon (6 sides) from the equilateral triangle. Can all regular polygons, with any number N of sides, be constructed with a ruler and compass?

The answer is no, but this was only understood in the 18th century, when it was proven that regular polygons with 7 and 13 sides cannot be constructed in this way. So what are the constructible values of N, that is, such that the regular polygon with N sides can be constructed using only a ruler and compass?

The problem attracted the attention of none other than the great Carl Friedrich Gauss. In 1796 he showed how to construct the regular heptadecagon (17 sides) using a ruler and compass. This was the discovery of which Gauss was most proud.

In his great work "Disquisitiones Arithmeticae" he went further, concluding that for a regular polygon to be constructible it is sufficient that the number N of sides be the product of a power of 2 by distinct Fermat prime numbers. He also stated that this condition would be sufficient, but this was only proven by the Frenchman Pierre Wantzel in 1837.

Pierre de Fermat calculated numbers of the form 1 plus 2 raised to the power of 2^ ⁿ for the values of n from 0 to 4, found that they were prime numbers, and believed that this would be true for all values of n. But, some years later, Leonhard Euler pointed out that Fermat's number with n=5 is not prime and, ironically, to this day no one has found any more besides the five original ones discovered by Fermat himself.

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