Ivars Peterson's MathTrek
May 27, 2002
Take a ribbon of paper, twist one end 180 degrees, and attach it to the other end. The resulting surface, called a Möbius strip, has only one side and one edge.
Since its discovery in the 19th century in a purely mathematical context, the Möbius strip has achieved a life of its own independent of mathematics--in the realms of magic, science, engineering, literature, music, film, art, and elsewhere. An object of continuing fascination, it has shown up in all sorts of unexpected settings: as monumental sculptures, knitted scarves and earbands, and glittering pendants; in designs on postage stamps and greeting cards; and as company logos and the ubiquitous symbol for recycling.
In 1982, chemists brought this remarkable topological form into the laboratory on a microscopic scale when they synthesized the first molecular Möbius strip. They formed the figure by joining the ends of a twisted, double-stranded strip of carbon and oxygen atoms.
Since then, researchers have coaxed a variety of molecules into Möbius strips, knots, and other curious structures. Now it's the turn of applied physicists, who have done the same thing with crystals.
In the May 23 Nature, Satoshi Tanda and coworkers at Hokkaido University in Japan describe a technique for twisting a crystalline ribbon of niobium selenide into a Möbius strip.
"It is surprising that a crystalline ribbon should adopt this exotic topology in view of its inherent rigidity, which would be expected to prevent it from either bending or twisting," the physicists remark.
Synthesized by chemical-vapor transport, fibrous crystals of niobium selenide (NbSe3) typically assume ribbon and whisker configurations. By tweaking the conditions normally used to grow these crystals from selenium and niobium powders, the researchers produced niobium selenide in the form of rings, Möbius strips, and figure-eights.
Such closed topologies occur when ribbon-shaped niobium selenide crystals adhere to droplets of liquid selenium, which act as spools around which the crystals are bent by surface tension. Growing along a droplet's equator to minimize bending energy, a crystal eventually meets its tail to form a seamless ring.
Apparently, Möbius crystals are produced when twisting accompanies bending or when the selenium droplet is rotating. The resulting Möbius crystals are about 50 micrometers in diameter and less than 1 micrometer in width.
The Japanese team suggests that the technique can be extended to a wide range of materials. The researchers have already obtained topological variants of tantalum compounds with selenium and sulfur. It's also possible to get different ring sizes by varying the liquid droplet diameter.
"Our crystal forms offer a new route to exploring topological effects in quantum mechanics, as well as to the construction of new [electronic or semiconductor] devices," they conclude.
Copyright 2002 by Ivars Peterson
Flapan, E. 2000. When Topology Meets Chemistry: A Topological Look at Molecular Chirality. Washington, D.C.: Mathematical Association of America.
Garmon, L. 1982. Möbius molecule: Synthesis with a twist. Science News 122(July 17):37.
Peterson, I. 2001. Möbius accordion. MAA Online (June 11).
______. 2000. Möbius at Fermilab. MAA Online (Sept. 4).
______. 2000. Möbius and his band. MAA Online (July 10).
Tanda, S., et al. 2002. A Möbius strip of single crystals. Nature 417(May 23):397-398.
Walba, D.M., R.M. Richards, and R.C. Haltiwanger. 1982. Total synthesis of the first molecular Möbius strip. Journal of the American Chemical Society 104(June 2):3219-3221.
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A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book Mathematical Treks: From Surreal Numbers to Magic Circles.