Soap Film and Minimal Surface


soap_filmThe geometry of soap films and soap bubbles

If we dip two wire rings into a solution of soapy water, then what is the surface formed by the soap film ? This question is known as the minimal surface problem, or the Plateau problem. This post proposes to discuss a simple solution based on variational calculus and elementary computations.

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The problem of Dido

Joseph Mallord William Turner, 1815

Dido building Carthage (William Turner, 1815)

HISTORY

Dido was the legendary founder of Carthage (Tunisia). When she arrived in 814BC on the coast of Tunisia, she asked for a piece of land. Her request was satisfied provided that the land could be encompassed by an ox-hide. With a remarkable mathematical intuition, she cut the ox-hide into a long thin strip and used it to encircle the land. This land became Carthage and Dido became the Queen.

PROBLEM

What is the closed curve which has the maximum area for a given perimeter ?

didoproblem

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