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A helicoid with a handle

For the first time in more than 200 years, a team of mathematicians from Rice, Stanford and Indiana universities has discovered a new shape of geometrical minimal shape. Their 'genus one helicoid' looks like a parking garage ramp or a curved soap film, but with a curved handle, like you can find on a coat hanger or a coffee mug.
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Written by Roland Piquepaille, Inactive on

For the first time in more than 200 years, a team of mathematicians from Rice, Stanford and Indiana universities has discovered a new shape of geometrical minimal shape. Their 'genus one helicoid' looks like a parking garage ramp or a curved soap film. But when it's untwisted, it looks like a flat plane with a curved handle, like you can find on a coat hanger or a coffee mug. But if this beautiful surface is a great example of geometric optimization, even the mathematicians recognize that there is no practical use today for a helicoid with a handle.

Here is the introduction of the Rice University news release.

It has been almost 230 years since French general and mathematician Jean Meusnier's study of soap films -- the same kind used by children today to blow bubbles -- led to one of the fundamental mathematical examples in geometric optimization. Meusnier showed that one of nature's simplest geometric figures -- an ordinary two-dimensional plane -- could be twisted infinitely into a helicoid, a shape that has the delicate balance everywhere of a soap film.

But now, mathematicians have found a new minimal surface.

David Hoffman of Stanford and Michael Wolf of Rice call the new surface a "genus one helicoid." From far away, the surface looks much like Meusnier's helicoid. However, when untwisted, the new shape differs from the flat plane of Meusnier's untwisted helicoid in a key way: It has a curved handle, much like the handle one might find on the flat lid of a kitchen pot.

Below are some pictures of such helicoids obtained with different sets of parameters (Credit: Rice University).

Helicoids with handles

What can we do with these new minimal surfaces? This is unclear.

Wolf said that while it is impossible to predict how the research will be applied to specific scientific problems, history has shown time and again that mathematical discoveries are almost invariably transmitted and transformed into useful solutions for society.
"I don't know of a practical use of a helicoid with a handle, but now I know that soap films are more flexible than they were once thought to be," Wolf said.

Anyway, the mathematical proof that this 'genus one helicoid' is a minimal surface has been published and is available from Rice University under the title "An embedded genus-one helicoid." Here is the abstract.

There exists a properly embedded minimal surface of genus one with one end. The end is asymptotic to the end of the helicoid. This genus one helicoid is constructed as the limit of a continuous one-parameter family of screw-motion invariant minimal surfaces -- also asymptotic to the helicoid -- that have genus equal to one in the quotient.

Here is a link to the very long paper (PDF format, 114 pages, 743 KB). But be warned: you might need to take some more math classes to understand it.

For more information about people or mathematical concepts discussed in this post, please visit these links from Wikipedia.

Sources: Rice University news release, October 31, 2005; and various NASA's web sites

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