23/10/2024

Folha: 'The Queen of Phoenicia and the Isoperimetric Problem'

Ruínas de Catargo, na Tunísia

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

The story is told by none other than the Roman poet Virgil (70–19 BC) in his epic poem "The Aeneid." Dido, queen of Tyre in Phoenicia, sails west with her followers, fleeing persecution by her brother, Pygmalion. Arriving in present-day Tunisia , she negotiates with the local king, Japon, for land for her people. In exchange for payment, Japon agrees to give her "all the land he can mark with an ox hide."

Cunningly, Dido orders the skin to be cut into very thin strips, sewn together to form a long cord, which she uses to delimit a semicircular area on the coast large enough to build a great city. Thus Carthage ("New Land" in the Tyrian dialect) is born, destined to become the most dangerous enemy Rome ever faced.

Impressed by the queen's intelligence, Japon asks her to marry him, but Dido refuses. Then the king builds a university, hoping to attract a future wife with equal mathematical talent. Just imagine, dear reader, how wonderful it would be if it weren't just a legend!

The mathematical question behind Dido's cunning—which we call the isoperimetric problem—was formulated in the 2nd century BC by the Greek Zenodorus (200–140 BC): "Of all plane figures with a given perimeter, which has the largest area?". The perimeter is the length of the figure's boundary: in the case of Carthage, it corresponds to the length of the cord.

Little is known of Zenodorus' life, and his original work has also been lost. However, commentaries by the Hellenistic mathematicians Pappus (290–350) and Theon (335–405), both from Alexandria, indicate that he 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 the most sides has the largest area. Notably, Zenodorus also studied the 3D version of the problem, showing that the sphere is the solid object with the largest volume among all those with a given surface area.

It wasn't just mathematicians who were interested in the isoperimetric problem. In "History of the Kings of Britain," a chronicle of King Arthur legends written in the 12th century by Geoffrey of Monmouth (1100–1155), the German Duke Hengist asks King Vortigern to pay him for his military services with land.

"Grant your servant only as much as can be encircled by a single rope within the land you have given me, so that I may build a high place where, if necessary, I may settle."

His wish granted, "Hengist immediately took the hide of a bull and forged it into a single strap, with which he encircled a rocky spot he had cleverly chosen, and within the space thus delimited, he began to build a castle."

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