Ponytails in motion. Credit: Mike Adams/Flickr via Creative Commons

BOSTON—At long last, one of the hairiest problems in modern physics has been solved. Researchers have devised a theoretical model to describe the shape of a ponytail.

A ponytail may look like a relatively simple object, but in truth it is a bundle of physical complexity. Multiple forces are in play. Each hair is elastic, with a random intrinsic curvature. And the average head of hair has 50,000 to 100,000 individual strands, according to Raymond Goldstein, a professor in the University of Cambridge's department of applied mathematics and theoretical physics.

Goldstein presented his ponytail research here at this week's meeting of the American Physical Society, and he and his colleagues at Unilever and the University of Warwick in the U.K. published a paper on their findings February 13 in Physical Review Letters.

Some of the major forces conspiring to shape a ponytail are elasticity, gravity, tension and pressure. That last property, Goldstein and his colleagues found, comes from the curvatures of individual fibers, from which the physicists derived a so-called equation of state for hair. A similar concept was applied more than 65 years ago in studies of the compressibility of wool.

In the numerical model that Goldstein and his colleagues devised, the push-and-pull of physical forces changes at various points along the ponytail. Near the base, swelling pressure and elasticity dominate. Beyond a few centimeters, the shape depends primarily on the pressure and the weight of the ponytail.

Beyond unveiling a theory of the ponytail, Goldstein and his colleagues also added a new term to the physics lexicon. They describe ponytail size by the "Rapunzel number," a unit equal to the total length of the ponytail in centimeters divided by five. Five centimeters, Goldstein said, is about the length scale below which gravity does not bend the hairs much.

To test their model, the researchers predicted ponytail shapes for various hair lengths and compared them to the real thing. They started with 25-centimeter ponytails and then cut off five centimeters at a time to measure how shape changes with length. "We take real ponytails and we trim them back," Goldstein said. "We reduce the Rapunzel number."

So how did the grand unified ponytail theory fare in its tonsorial test? "The answer is, we do a pretty good job," Goldstein reported.