Mathematicians describe tendril perversion

by M.N. Jensen

Tendrils of Bryonia dioica (top), a plant in the squash family, exhibit perversion in this 1875 drawing from J. Sachs' Text-book of Botany. The bottom half of the tendril coils one way, while the top half coils in the opposite direction. Close-up of a telephone cord (bottom) shows the junction of the right-handed and left-handed coils.

The twisting tendrils of climbing plants have intrigued biologists for well over a century. A tendril can coil first in one direction and then in the opposite direction. This phenomenon, called perversion, also shows up in kinky telephone cords. Now, mathematicians have shown that such reversals of direction result from the curviness inherent in tendrils and cords. By solving mathematical equations that describe the idealized behavior of thin, flexible rods such as telephone cables or licorice sticks, Alain Goriely of the Université Libre of Brussels and Michael Tabor of the University of Arizona in Tucson show that putting such a rod under tension can spontaneously generate left-handed and right-handed twisting behavior. "We are modeling the vine tendril as a thin elastic rod, and our results suggest that's not a bad physical representation," says Tabor. The findings appear in the Feb. 16 Physical Review Letters.

"I am repeatedly amazed at the range of applications of the theory of the deformation of elastic rods, from DNA to climbing plants," says Ellis H. Dill of Rutgers University in Piscataway, N.J. "They've shown that this interesting phenomenon, climbing plants, is explainable by classical physics."

The rods behave this way "because of the elastic nature of the substance, independent of any [specific] molecular structure," he adds.

The key to the reversal of twist, Tabor says, is a property called intrinsic curvature. "If you take your phone cord and stretch it out, it's straight," he says. "If you let it go, it has loops. So you would say it naturally wants to have loops."

Goriely and Tabor figured out the importance of intrinsic curvature by finding solutions to Kirchoff's theory of thin elastic rods, a 100-year-old set of equations. Solving the Kirchoff equations was very difficult, however. To do so, the two mathematicians developed some new analytical techniques and used computer algebra (SN: 3/22/97, p. 176).

The results show how to build what Tabor calls a twistless spring -- a spring that starts by coiling one way and then reverses, and thus has a net twist of zero. Both vine tendrils and telephone cords form such springs.

"Here's the funny thing," Tabor says. "The vine is locked on two ends. It has no twist in it, yet it would really like to be like a spring, to absorb motion. . . . It winds up one way, and then it changes direction and winds the other way. The right-handed twist and the left-handed twist cancel."

The two different directions of twist represent two different solutions to the Kirchoff equations. Each solution describes a different state, and these states alternate back and forth.

Cycling among different states has been predicted mathematically for a wide range of systems that have suitable symmetry. For example, animals can, in principle, switch between right-footed and left-footed gaits. Such cycling has been observed only rarely in nature.

"In systems with symmetry, it's one of the things you expect to have happen," says Martin Golubitsky of the University of Houston in Texas. "I have been sitting around with colleagues for the past 2 years muttering about why we don't see these cycles, because mathematically we know they're there.

"I am happy to see it come about in this really pretty physical manifestation," he adds. "We learn something about the mathematics by seeing how it is realized in the physical system."

Alain Goriely
Université Libre de Bruxelles
Département de Mathématique
CP 218/1
1050 Brussels
Belgium

Michael Tabor
University of Arizona
Program in Applied Mathematics
Tucson, AZ 85721

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