# Isoperimetric problem (Dido’s Problem)

## History

Dido (or Elissa, depending on the reference) is the legendary founder and first Queen of Carthage. Her accounts are mentioned by the historian Justin and later as a part of Virgil’s “Aeneid” poem. In this story, Elissa is said to have escaped Tyre with groups of her brother Pygmalion’s attendants and senators. Eventually Elissa’s party arrived on the coast of Northern Africa where Elissa had asked the local inhabitants for a small area of land for a refuge. The isoparametric problem came into existence. Elissa asked for only as much land as could be encompassed by an oxhide. She used a stroke of mathematical genius, cutting the hide into long, thin strips in order to encircle the entirety of the nearby hill. This land became what is Carthage, and Elissa (or Dido) became the Queen of the city.

The history of Elissa is mentioned by Justin, however, the problem is formulated from a passage in Virgil’s “Aeneid”:

The Kingdom you see is Carthage, the Tyrians, the town of Agenor;
But the country around is Libya, no folk to meet in war.
Dido, who left the city of Tyre to escape her brother,
Rules here – a long and labyrinthine tale of wrong
Is hers, but I will touch on its salient points in order…
Dido, in great disquiet, organized her friends for escape.
They met together, all those who harshly hated the tyrant
Or keenly feared him: they seized some ships which chanced to be ready…
They came to this spot, where today you can behold the mighty
Battlements and rising citadel of New Carthage,
And purchased a site, which was named “Bull’s Hide” after the bargain
By which they should get as much land as they could enclose with a bull’s hide. [1]

## Mathematical Formulation

Knowing the history helps us to see how this might be useful, but it is not formalized in any mathematical sense. If we think of the problem, the problem becomes similar to what is known in mathematics as the isoperimetric problem. In Dido’s problem, Dido cuts the pieces of the ox-hide into strips, effectively using the area of the hide as the perimeter. From this point, the goal of the Dido problem then is to find the closed curve that has maximal area for a given perimeter. The problem is at this point, the isoperimetric problem.

Isoperimetric, literally means “having the same perimeter”, therefore, we know that the perimeters are fixed and our goals is to simply maximize the area. The isoperimetric problem requires the use of the isoperimetric inequality (and quotient) where if we think planar geometry, we can define a figure to have an area A and a perimeter p. The quotient then is [2]:

$Q\equiv \frac{4\pi A}{p^2} \leq 1$

This quotient is derived from the ratio of the curve area to the area of a circle ($A = \pi r_A^2$) with the same perimeter as the curve ($p=2 \pi r_p$). We can therefore derive the quotient:

$\begin{matrix} Q & \equiv & \frac{r_A}{r_p^2}\\ & = & \frac{\left ( \frac{A}{\pi}\right )}{ \left ( \frac{p}{2 \pi} \right )^2} \\ & = & \frac{4 \pi A}{p^2} \end{matrix}$

Mathematics tells us that this inequality holds for the shape of a circle only. Mathematical Proofs for this concept representing a can be found at [3-6]

## Usefulness in Computer Science

In combinatorics and computer science, the use of Isoperimetric inequalities (the basis of the Dido problem) plays a role in designing robust computer networks, several applications in complexity theory, and error-correcting codes, through the use of expander graphs.[7] Expander graphs are a sparse graph with strong connectivity properties. The isoperimetric problems in graph theory (described on page 470 of [7]) then have usefulness in graph optimization problems for expander graphs.

There are perhaps many other applications of the Isoperimetric problem and the Dido problem, as the Dido problem is in itself an optimization of maximal area given a perimeter.

## Works Cited:

[1] Virgil. Trans. by C.D. Lewis. Book 1, lines 307-372 in The Aeneid. New York: Doubleday, pp 22-23. 1953.
[2] Osserman, R. “Isoperimetric Inequalities.” Appendix 3, Sec 3 in A Survey of Minimal Surfaces. New York: Dover, pp. 147-148, 1986.
[3] Bonnesen, Les Problèmes des Isopérîmètres et des Isépîphanes. Paris: Gauthier-Villars, pp 59-61, 1929.
[4] Bonnesen, T. and Fenchel, W., Theorie der Convexen Körper. Chelsea Publishing, New York, S.111-112, 1948.
[5] Magnani, C. “The Problem of Dido” Internet: http://mathematicalgarden.wordpress.com/2008/12/21/the-problem-of-dido/, December 2008 [September 5, 2011].
[6] Luthy, P. “Two Cute Proofs of the Isoperimetric Inequality” Internet: http://cornellmath.wordpress.com/2008/05/16/two-cute-proofs-of-the-isoperimetric-inequality/, May 16, 2008 [September 5, 2011].
[7]Hoory, S., Linial, N., and Wigderson, A. “Expander Graphs and Their Applications” Bulletin of the American Mathematical Society 43 (4): 439-561, 2006. dpi: 10.1090/S0273-0979-06-01126-8.