A Spiral Proof

Researchers have proven the existence of a structure that until now, wasn't even thought to be mathematically possible.

The new shape [bottom image] looks like a traditional helicoid [top image]—the spiral shape that results if a flat plane is twisted an infinite number of times—but with one exception: in the middle, there is a coffee-mug-like "handle."

The new structure is called a "genus one helicoid," and its existence has been debated since the early 1990's. Recently, however, two mathematicians, Mattias Weber of Indiana University and David Hoffman of Stanford University, developed a proof that such structures exist.

The genus one helicoids is an example of what mathematicians call a minimal surface, which is formed when the pressure on both sides of a surface are the same.

"For example, when you dip a bent coat hanger into soapy water, the soap bubble that forms on the hanger is a minimal surface," Weber explained. Depending on the shape of the coat hanger, the soap bubble can take on various shapes, but in every case the bubble is trying to take up the least amount of surface area possible.

Minimal surfaces occur in nature and are extremely stable as physical objects. They are proving to be important at the molecular level because they often occur at the places where two substances meet. Knowing what these interfaces look like can help researchers determine what the chemical properties of new mixtures will be.

The finding was detailed in the Nov. 15 issue of the journal for the Proceedings of the National Academy of Sciences.

--Ker Than

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Credit: : Indiana University