Did you know that the shape enclosing the largest area for a given perimeter is the circle? On this fascinating journey through one of the oldest problems in mathematics, discover how Queen Dido used her ingenuity to found Carthage and how mathematicians, from the Greeks to the modern era, have tried to prove this surprising intuition. Learn about the isoperimetric problem, its impact on literature and science, and how it remains an active field of research today. Don’t miss this story full of legends, proofs, and a touch of mathematical magic! ✨

This seemingly simple problem, the isoperimetric problem, took over 2000 years to be proven.

To find its origin, we go back to Virgil, who in the Aeneid recounts how the ingenious Queen Dido founded Carthage with the help of geometry. King Jarbas had promised her all the land along the coastline that she could encompass with an oxhide. The sovereign made this "generous" offer intending to give her the minimum land, but Dido maximized it with cunning. She cut the hide into very thin strips to make a long thread with which she drew... Indeed, a semicircle! In that space, she founded the famous Phoenician city.

The Riemann hypothesis would have loved to begin in such a beautiful way, instead of as a comment in a brief article. Just as this hypothesis generates great interest today, the isoperimetric problem did from antiquity to the 19th century.

In ancient Greece, the problem is associated with Zenodorus. His treatise "On Isoperimetric Figures" is lost and only known through references. Zenodorus concluded that the area of the circle is greater than that of any polygon with the same perimeter, but his proof contained an error that was not corrected until much later.

The list of mathematicians who attempted to solve isoperimetric problems is extensive: Galileo, the Bernoulli brothers, Euler, Legendre, and more.

But it is the Swiss mathematician Jakob Steiner who is most intimately linked to the isoperimetric problem. On one hand, he corrected Zenodorus' error, and on the other, he provided a very elegant and intelligent geometric proof of the problem. However, it was incomplete because he did not prove the existence of the solution! This brought him quite a few troubles.

We had to wait until Weierstrass found a complete and rigorous solution using calculus of variations.

The isoperimetric problem generalizes in space with the solution being the sphere, which is equivalent to saying that for a fixed volume, the minimal surface enclosing it is a sphere. So, when it's cold, our dog, who has a fixed volume, minimizes her surface contact with the exterior by curling up into a sphere to lose less heat. (The isoperimetric problem explains why dogs and cats sleep curled up).

Chapter Index:
00:00 Introduction to the isoperimetric problem
01:08 The legend of Queen Dido
03:22 Mathematicians who have worked on this problem: Galileo, Euler, the Bernoulli brothers, Gauss, Legendre, Steiner, Weierstrass, Schmidt...
05:04 Two preliminary lemmas
07:21 Steiner's proof
12:16 The importance of existence
12:47 Perron's "quoted" theorem
14:04 Calculus of variations
15:16 Generalization and dualization (Our dog applies the theorem)

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