Dido's Problem

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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 Dido's Problem.

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

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