Raymond E. Goldstein, a professor of complex physical systems at the University of Cambridge in England, does not have a ponytail, but he has been pondering the physics of the hairstyle for a couple of years.
He and two other physicists have been trying to determine whether the shape of a ponytail can be deduced from the properties of a single hair. After all, a head with 100,000 strands is a complex physical system, as anyone with a copious coiffure can attest.
And it turns out that there is a simple theory. The crucial characteristics are elasticity, density and curliness, which essentially tell how springy a piece of hair is, plus the length of the ponytail. The researchers came up with a simple formula that describes the ponytail shape when the hair is bundled together.
They called it the Rapunzel number. “We couldn’t resist,” Dr. Goldstein said.
A short ponytail of springy hair, characterized by a low Rapunzel number, fans outward. A long ponytail with a high Rapunzel number hangs down, as the pull of gravity overwhelms the springiness. “I think we were surprised about the simplicity of this,” Dr. Goldstein said.
Dr. Goldstein, with Robin C. Ball, a physicist at the University of Warwick in England, and Patrick B. Warren, a researcher at Unilever, report their findings in Monday’s issue of the journal Physical Review Letters, adding to the long scientific literature on hair.
Five centuries ago, Leonardo da Vinci sketched in his notebooks an observation that the waviness of hair resembled eddies in the flow of rivers. More recently, Joseph B. Keller, a Stanford mathematician, figured out why the ponytails of joggers sway from side to side, instead of up and down with the jogger’s head. He found that the up-and-down motion was unstable, and that the ponytail could not sway forward and backward because the jogger’s head was in the way. Thus, any slight jostling caused the up-and-down motion to become side-to-side swaying.
A few years ago, Unilever, the multinational corporation whose products include soaps and shampoos, approached Dr. Goldstein about collaborating on research on the basic properties of hair, including tangling and the shapes of individual strands.
“Somehow,” Dr. Goldstein recalled, “a bunch of balding, middle-aged men sitting around a table came up with the idea that the ponytail was the embodiment of all this interesting physics.”
Dominic Tildesley, a vice president for research and development at Unilever, said he hoped that the work could help the company create better hair products. It could also aid computer animators trying to depict the hair movements of their characters. Dr. Goldstein said the findings could also be applied to bundles of other long filaments, including fiberglass and wool.
Dr. Goldstein said his follow-up research would combine his findings with Dr. Keller’s for a better understanding of ponytails.
Hair News Network