# Mathematicians solve old problem that may have new applications

##### December 25, 2005

A twisted soap bubble with a handle? If you find that hard to visualize, it's understandable. Experts had thought for more than 200 years that such a structure was not even mathematically possible. But no longer.

In a paper published in the Nov. 15 issue of Proceedings of the National Academy of Sciences, mathematicians Matthias Weber of Indiana University, David Hoffman of Stanford University, and Michael Wolf of Rice University presented a proof of the existence of a new minimal surface they call a genus one helicoid.

"This proof tells us that our intuition was not quite right about what is possible and what is not possible," Weber said. "Probably one reason it was not discovered sooner is that no one imagined that something like this could exist."

A helicoid is what results when one of the simplest shapes -- a flat plane -- is twisted an infinite number of times. If the helicoid is vertical, its shape resembles a spiral parking ramp.

The new surface looks much like this traditional helicoid with an extra feature: a handle such as one finds on a coffee mug. It turns out that on one "floor" of the parking ramp there is an additional column -- the handle.

All minimal surfaces have something important in common: a minimal surface area.

"A minimal surface is formed when the pressure on both sides of a surface is the same," Weber explained. "'For example, when you dip a bent coat hanger into soapy water, the soap bubble that forms on the hanger is a minimal surface." These soap bubbles can have various shapes, depending on the shape of the coat hanger, but in every case the bubble is trying to minimize surface tension, he said. This happens when the bubble has the smallest possible surface area.

At every point, a minimal surface is either flat or shaped like a saddle or a potato chip.

Minimal surfaces are proving to be important at the molecular level. "Minimal surfaces actually occur in nature at the nanoscale as interfaces between certain substances," Weber said. An example is some copolymers that are plastics used to make new kinds of fabrics. When these copolymers are mixed, there are interfaces between them that are minimal surfaces. Knowing what these interfaces look like can help in determining what the chemical properties of the mixture will be.

Minimal surfaces are extremely stable as physical objects, Weber pointed out, and this can be an advantage in many kinds of structures. He has heard from architects who have seen computerized illustrations of some of his minimal surfaces and are intrigued by the possibility of adapting them to structures, both interior and exterior. He has exchanged information about minimal surfaces with some architects and is exploring ways to collaborate with them.

Calendars are another use for this work, highlighting the aesthetic qualities of minimal surfaces. These aesthetic qualities are on vivid display in Weber's computer gallery of minimal surfaces at www.indiana.edu/~minimal/gallery/index/index.html , which shows minimal-surface objects set in imaginary landscapes.

"The images in the gallery are not intended as illustrations of mathematical facts," Weber said. "They more than fulfill their purpose if people see them and can feel some of the intriguing enchantment that a mathematician feels when exploring the mathematical objects."

In a second gallery at www.indiana.edu/~minimal/archive/index.html the pictures do illustrate mathematical facts. "This is the most comprehensive collection of minimal surfaces available," Weber said. "Users can download programs that recreate the surfaces, allowing them to conduct numerical and visual experiments."

The mathematicians' complete proof is more than 100 pages long. The abstract of their report in Proceedings of the National Academy of Sciences is available at www.pnas.org/cgi/content/abstract/102/46/16566 .

Source: Indiana University

Explore further: Superomniphobic tape adheres to any surface

## Related Stories

#### Superomniphobic tape adheres to any surface

October 20, 2016

Arun Kota, assistant professor of mechanical engineering at Colorado State University, has made a superomniphobic tape that, when adhered to any surface, gives the surface liquid-repelling properties. This recent breakthrough ...

#### World first porous acupuncture needles enhance therapeutic properties

October 17, 2016

A Daegu Gyeongbuk Institute of Science and Technology research team led by Professor Su-Il In, who developed acupuncture needles combined with nanotechnology, was recognized as the world's first application of this technology. ...

#### Russian scientists' method allows forecast hurricanes more precisely

October 18, 2016

Among the most dangerous phenomena on the planet, hurricanes are giant atmospheric vortexes with air pressure subsiding to the centre and a very high speed of the air flow. They appear in the middle latitudes because of the ...

#### Swiss doctors report success of using cells from the nose to repair damaged knee joints

October 21, 2016

Writing in The Lancet, Swiss doctors report that cartilage cells harvested from patients' own noses have been used to successfully produce cartilage transplants for the treatment of the knees of 10 adults (aged 18-55 years) ...

#### Getting maximum profit, minimal pollution

October 12, 2016

There are more chickens in the United States than people in the entire world. Raising huge numbers of chickens generates large quantities of waste. This waste includes feces, feather, and bedding materialsâ€”collectively ...

October 12, 2016

In a paper this week for the journal Nature Energy, a Los Alamos National Laboratory research team demonstrates an important step in taking quantum dot, solar-powered windows from the laboratory to the construction site by ...

## Recommended for you

#### Young stellar system caught in act of forming close multiples

October 26, 2016

For the first time, astronomers have seen a dusty disk of material around a young star fragmenting into a multiple-star system. Scientists had suspected such a process, caused by gravitational instability, was at work, but ...

#### Nickel-78 is a 'doubly magic' isotope, supercomputing calculations confirm

October 26, 2016

For many of us, the term "doubly magic" may evoke images of Penn & Teller. However, for nuclear physicists, it describes atomic nuclei that have greater stability than their neighbors thanks to having shells that are fully ...

#### Universal clamping protein stabilizes folded proteins: New insight into how the chaperone protein Hsp70 works

October 26, 2016

On October 26th Nature will publish a study that overturns the decades-old textbook model of action for a protein that is central for many processes in living cells. Researchers at the FOM Institute AMOLF and the University ...

#### Astronomers use observations of a gravitationally lensed galaxy to measure the properties of the early universe

October 26, 2016

Although the universe started out with a bang it quickly evolved to a relatively cool, dark place. After a few hundred thousand years the lights came back on and scientists are still trying to figure out why.

#### For the first time, brain surface stimulation provides 'touch' feedback to direct movement

October 26, 2016

In the quest to restore movement to people with spinal cord injuries, researchers have focused on getting brain signals to disconnected nerves and muscles that no longer receive messages that would spur them to move.

#### Researchers invent 'perfect' soap molecule that is better for the environment

October 26, 2016

A team of researchers, led by the University of Minnesota, has invented a new soap molecule made from renewable sources that could dramatically reduce the number of chemicals in cleaning products and their impact on the environment.