Dido's Problem

Classic Problem

The new City of Carthage is to be founded.

Its area is to be determined by enclosing as much land as possible within a line whose length is fixed.

What shape should it be to make sure the area is a maximum?


Solution

The classical answer is that the enclosure was in the shape of a semicircle whose diameter is the coastline.

However, it is clear that this depends upon the shape of the coast, and a better solution may be to cut off a peninsula.


Proof

Let the shoreline be assumed to be a straight line.

Imagine the enclosure takes some geometric figure $S$.

Let $S$ be reflected in the shoreline.

Then the entire geometric figure formed by $S$ along with its reflection $S'$ encloses the largest area for double the length of the boundary line.

This largest area is a circle.


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Historical Note

There are several more or less romantic legends of the origin of .

One of the more prosaic is that the founders were granted as much land as a man could plough a furrow round it in a day, given that a ploughman works at a constant rate.


A more fanciful one concerns Queen Dido, who was given a bull's hide, and told she could take as much land for her new city as she could enclose. According to the legend, she cut it (or arranged to have it cut -- she was a queen, after all) into one long strip of leather.

In the words of Virgil:

So they reached the place where you will now behold mighty walls and the rising towers of the new town of Carthage;
and they bought a plot of ground named Byrsa ...
for they were to have as much as they could enclose within a bull's hide.
-- Virgil, Aeneid, Book $\text I$, ll. 360 -- 70


In both cases, the resulting length was used to measure out a semicircle whose diameter formed the coastline.


The legend of the bull's hide is repeated throughout history in the contexts of the founding of several cities.


Sources

  • 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VIII}$: Nature or Nurture?
  • 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Dido's problem
  • 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The Area Enclosed Against The Seashore: $30$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Dido's problem
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Dido's problem
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