A Helix with a Handle
Dip a loop of wire into a soapy solution, and the film that covers
the loop will be what mathematicians call a minimal surface. The
soap forms such a shape because it minimizes surface tension. At any
point, a minimal surface is maximally curved in one direction and
minimally curved in the opposite direction, but the amount of
curvature in each direction is exactly the same. As a result, each
point on the surface is either a flat plane or a saddle shape, never
a sharp peak or valley. But a minimal surface doesn't have to be
flat or simple overall: A plane can be twisted into a parking-ramp
shape called a helicoid, which mathematicians proved over two
centuries ago is also a minimal surface.
Mathematicians have proved the existence of a class of minimal
surfaces that cannot be embodied by soap bubbles but can be
visualized by computer simulation. This surface, called a genus-one
helicoid, is a variation on a standard helicoid, but there is a
tunnel through the deck of the parking-ramp spiral. When untwisted,
this surface looks like a flat sheet with a coffee-mug-handle shape
grafted onto it. "Think of a torus, like an inner tube,"
says Matthias Weber of Indiana University. "Now imagine that
you puncture the torus. This results in a surface that can be
stretched and deformed into the genus-one helicoid. I think that's a
real mind bender."
As they reported in the November 15, 2005, issue of the
Proceedings of the National Academy of Sciences, Weber,
David Hoffman of Stanford University and Michael Wolf of Rice
University have proven that such shapes, whether they have one or an
infinite number of handles, are indeed minimal surfaces that can go
on forever in all directions and never fold back to intersect themselves.
Over a decade ago, Hoffman, with Fusheng Wei, then of the University
of Massachusetts at Amherst, and Hermann Karcher of the University
of Bonn in Germany, had created computer simulations of such handled
helicoids, but an airtight demonstration of minimal surfacehood
eluded them. "Computer graphics programs enabled us to
visualize these surfaces, but we couldn't bring them back into the
mathematical fold," says Hoffman. "I think the information
about how to solve this problem was lurking in the pictures all the
time, but we just had to think about it for a long time and have the
theory catch up with the evidence we had." Catching up can be
hard to do: The mathematical proof takes up more than 100 pages.
An advanced understanding of minimal surfaces could be relevant to
materials science; for instance, some compound polymers, such as
Kevlar, have interfaces between molecules that are approximately
minimal surfaces, the shape of which can influence the chemical
properties of the material.
As mathematicians, Weber and his colleagues are most excited about a
potentially large, new class of minimal surfaces that have not been
found in nature and which no investigators had imagined could exist
until recently. "It's easy to come up with one new example of a
minimal surface, but this one is of a very different nature than
others that have been found before," Weber said. "So it's
opened a new field within the theory of minimal surfaces."