minimal surface

Costa's minimal surface

A minimal surface is a surface that, bounded by a given closed curve or curves, has the smallest
possible area. A minimal surface has a mean curvature of zero. Finding and classifying minimal surfaces and proving that certain
surfaces are minimal have been major mathematical problems for over 200
years. If the closed curve is planar then the solution is trivial; for example,
the minimal surface bounded by a circle is just a disk. But the problem
becomes much more difficult if the bounding curve is nonplanar –
in other words, is allowed to move up and down in the third dimension. The
first nontrivial examples of minimal surfaces, the catenoid and the helicoid, were discovered by the
French geometer and engineer Jean Meusnier (1754–1793) in 1776, but
there was then a gap of almost 60 years before the German Heinrich Scherk
found some more. In 1873 the Belgian physicist Joseph Plateau carried out
some experiments which led him to conjecture that soap bubbles and soap films always form minimal surfaces. Proving mathematically this
was true became known as Plateau's
problem. Most minimal surfaces are extremely hard to construct and visualize,
in part because the majority of them are selfintersecting. However, the
development of high performance computer graphics has provided mathematicians
with a powerful tool and the last couple of decades of seen a huge increase
in the number of such surfaces that have been defined and investigated.
Related category
• SOLIDS
AND SURFACES
