The engaging isoperimetric problem

Virgil, an ancient Roman poet, composed a Latin epic poem known as Aeneid. It tells the legendary story of the ancient Phoenician queen Dido, who escapes from her brother to reach the North African coast of modern-day Tunisia. The local people there agree to sell her an amount of land that can be bounded by the hide of a bull. To their surprise, Dido cuts the hide into long, narrow strips and proceeds to arrange them in such a way that they enclose the maximum area with the given straight-line seashore. She apparently knows, correctly, that this would be in the shape of a semicircle. In this way, she is able to enclose a large enough area to establish a kingdom, thus finding the ancient city of Carthage along with a famous problem in mathematics.

The isoperimetric problem seeks the (closed) shape that, for a given perimeter, maximises the area enclosed. Dido’s problem was closely related to this, just that it had the constraint that one part of the shape must be a straight line (the seashore). From the semicircle solution to Dido’s problem, it seems that the solution to the isoperimetric problem should be a circle, which is in fact true. This result can be stated as an β€˜inequality’, called the isoperimetric inequality. For any shape with perimeter L and enclosed area A, L2 β‰₯ 4Ο€A holds. The equality occurs only when the shape is a circle. Try checking this by substituting L for the perimeter of a circle and A for its area.

Contributions towards proving this result came from as early as ancient Greece, where Zenodorus proved some significant but incomplete results. In the 19th century, Steiner made substantial progress towards a rigorous mathematical proof, which was ultimately completed by Weierstrass.

We can observe the isoperimetric inequality in nature in many ways. One such demonstration involves dipping a frame (of any shape) with a loop of string attached to it into a soap solution. This causes a soap film to form and cover the entire frame as well as the loop of string. If we now pop only the soap film inside the loop, then the loop takes a circular shape. This is because the energy of the soap film in the frame is proportional to the area of the film. To minimise energy it must minimise its area, which means it must maximise the area inside the string loop (since that area is not covered by the film) which is achieved by making the loop circular.

The isoperimetric inequality can be generalised to dimensions greater than two as well, with length and area being replaced by appropriate quantities. From the ancient Carthage to the modern world, the isoperimetric inequality has stood its ground when it comes to being engaging, and continues to do so.

By Gaurav Pundir

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