30/10/2024

In Folha, Viana discusses the isoperimetric problem.

Jacob e Johann Bernoulli

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

The problem dates back at least to the Greek Zenodorus (200–140 BC), who, in the 2nd century BC, already asked: "Of all plane figures with a given perimeter, which has the largest area?". His original text, "On Isoperimetric Figures", has been lost. But not before it had been read by Pappus (290–350) and Theon (335–405) of Alexandria, according to whom Zenodorus proved that the circle has a larger area than any regular polygon with the same perimeter and that among these regular polygons, the one with more sides has the largest area.

Pappus's commentaries, published at the end of the 3rd century, are particularly interesting. Volume 5 of his "Mathematical Collections" begins with a preface entitled "The sagacity of bees," in which he attributes the almost perfect hexagonal geometry of the beehive to "a certain mathematical premeditation" on the part of the insects. He points out that this geometry helps to store as much honey as possible in the available space and uses this observation to explain why the isoperimetric problem deserves to be studied, "since we have even more wisdom than bees."

However, rigorously proving that the circle is indeed the curve with the largest area among all curves with a fixed perimeter was beyond the reach of ancient mathematicians, as it required mathematical advancements, such as the discovery of calculus, which would only occur from the 17th century onwards.

The first significant progress was made by the brothers Jacob (1654–1705) and Johann (1667–1748) Bernoulli in the 1690s. The two had an intense rivalry that grew over the years until it turned into mutual hatred. The fact that both worked on the isoperimetric problem, using similar ideas, contributed greatly to this aggravation. Johann even announced a solution, but it was wrong, and Jacob took the opportunity to mock his brother. Nevertheless, their work on this and other similar problems marks the beginning of a new mathematical discipline: the calculus of variations.

Leonhard Euler (1707–1783), who studied with Johann Bernoulli, was also interested in the isoperimetric problem. In a work published in 1744, he introduced new techniques to approach the problem, coming close to proving that the circle is indeed the solution to the question, that is, the only curve that maximizes the area of the region bounded by it for each fixed perimeter.

The methods that Euler presented were improved by Joseph-Louis Lagrange (1736–1813) and are the basis for the great progress that the calculus of variations has had since then.

The complete solution to the isoperimetric problem was only achieved in the 19th century, with important contributions from several mathematicians, such as Karl Weierstrass, Bernhard Riemann, Jacob Steiner, John von Neumann, Constantin Carathéodory, and others.

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