Forum for Science, Industry and Business

Sponsored by:     3M 
Search our Site:

 

Getting a handle on minimal surfaces

21.12.2005


The top image is a traditional helicoid. The bottom one has a hole in it that would become a handle if the shape were completely untwisted into a flat surface.


IU mathematician helps solve old problem that may have new applications

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 http://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 http://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."

A detailed article about the helicoid discovered by Weber and his co-workers can be found in the Dec. 17 issue of Science News at http://www.sciencenews.org/articles/20051217/bob9.asp.

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 http://www.pnas.org/cgi/content/abstract/102/46/16566.

Weber can be reached at 812-855-8724 or matweber@indiana.edu.

Matthias Weber | EurekAlert!
Further information:
http://www.indiana.edu

More articles from Physics and Astronomy:

nachricht A 100-year-old physics problem has been solved at EPFL
23.06.2017 | Ecole Polytechnique Fédérale de Lausanne

nachricht Quantum thermometer or optical refrigerator?
23.06.2017 | National Institute of Standards and Technology (NIST)

All articles from Physics and Astronomy >>>

The most recent press releases about innovation >>>

Die letzten 5 Focus-News des innovations-reports im Überblick:

Im Focus: Can we see monkeys from space? Emerging technologies to map biodiversity

An international team of scientists has proposed a new multi-disciplinary approach in which an array of new technologies will allow us to map biodiversity and the risks that wildlife is facing at the scale of whole landscapes. The findings are published in Nature Ecology and Evolution. This international research is led by the Kunming Institute of Zoology from China, University of East Anglia, University of Leicester and the Leibniz Institute for Zoo and Wildlife Research.

Using a combination of satellite and ground data, the team proposes that it is now possible to map biodiversity with an accuracy that has not been previously...

Im Focus: Climate satellite: Tracking methane with robust laser technology

Heatwaves in the Arctic, longer periods of vegetation in Europe, severe floods in West Africa – starting in 2021, scientists want to explore the emissions of the greenhouse gas methane with the German-French satellite MERLIN. This is made possible by a new robust laser system of the Fraunhofer Institute for Laser Technology ILT in Aachen, which achieves unprecedented measurement accuracy.

Methane is primarily the result of the decomposition of organic matter. The gas has a 25 times greater warming potential than carbon dioxide, but is not as...

Im Focus: How protons move through a fuel cell

Hydrogen is regarded as the energy source of the future: It is produced with solar power and can be used to generate heat and electricity in fuel cells. Empa researchers have now succeeded in decoding the movement of hydrogen ions in crystals – a key step towards more efficient energy conversion in the hydrogen industry of tomorrow.

As charge carriers, electrons and ions play the leading role in electrochemical energy storage devices and converters such as batteries and fuel cells. Proton...

Im Focus: A unique data centre for cosmological simulations

Scientists from the Excellence Cluster Universe at the Ludwig-Maximilians-Universität Munich have establised "Cosmowebportal", a unique data centre for cosmological simulations located at the Leibniz Supercomputing Centre (LRZ) of the Bavarian Academy of Sciences. The complete results of a series of large hydrodynamical cosmological simulations are available, with data volumes typically exceeding several hundred terabytes. Scientists worldwide can interactively explore these complex simulations via a web interface and directly access the results.

With current telescopes, scientists can observe our Universe’s galaxies and galaxy clusters and their distribution along an invisible cosmic web. From the...

Im Focus: Scientists develop molecular thermometer for contactless measurement using infrared light

Temperature measurements possible even on the smallest scale / Molecular ruby for use in material sciences, biology, and medicine

Chemists at Johannes Gutenberg University Mainz (JGU) in cooperation with researchers of the German Federal Institute for Materials Research and Testing (BAM)...

All Focus news of the innovation-report >>>

Anzeige

Anzeige

Event News

Plants are networkers

19.06.2017 | Event News

Digital Survival Training for Executives

13.06.2017 | Event News

Global Learning Council Summit 2017

13.06.2017 | Event News

 
Latest News

Quantum thermometer or optical refrigerator?

23.06.2017 | Physics and Astronomy

A 100-year-old physics problem has been solved at EPFL

23.06.2017 | Physics and Astronomy

Equipping form with function

23.06.2017 | Information Technology

VideoLinks
B2B-VideoLinks
More VideoLinks >>>