Researchers Crack The Ponytail Code
February 13, 2012

Researchers Crack The Ponytail Code

British scientists reported Friday they have cracked the mathematical conundrum behind the shape of hair that has perplexed humanity since Leonardo da Vinci first pondered it some 500 years ago.

The “Rapunzel Number,” a mathematical equation devised by scientists from the University of Cambridge and the University of Warwick, has helped quantify the curliness of human hair and can be used to predict the shape of any ponytail.

Cambridge Professor Raymond Goldstein and colleagues took account of the stiffness of individual hairs, the effects of gravity and the average waviness of human hair to produce the formula.

“That determines whether the ponytail looks like a fan or whether it arcs over and becomes nearly vertical at the bottom,” Goldstein told Li-mei Hoang of Reuters in a telephone interview.

The research also took into account how a bundle of hair is swelled by the outward pressure which arises from collisions between component hairs.

“It´s a remarkably simple equation,” explained Goldstein, who is the Schlumberger Professor of Complex Physical Systems at Cambridge´s Department of Applied Mathematics and Theoretical Physics, in a recent press release. “Our findings extend some central paradigms in statistical physics and show how they can be used to solve” the riddles of human hair.

“To be able to reduce this problem to a very simple mathematical form which speaks immediately to the way in which the random curliness of hair swells a ponytail is deeply satisfying. Physicists aim to find simplicity out of complexity, and this is a case in point,” he added.

“We imagine that at least half of the population has direct experience with the properties of ponytails, and we all have likely wondered about the fluffiness of hair,” said Goldstein, whose research was partially funded by the Schlumberger Chair Fund.

The study, published in the journal Physical Review Letters, provides the first quantitative understanding of the distribution of hairs in a ponytail.

Explaining their findings, the researchers 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,” said Goldstein, who will be presenting the research -- along with coauthors Robin C Ball, a physicist at Warwick, and Patrick B Warren, a researcher at Unilever -- at a meeting of the American Physical Society in Boston on February 28, 2012.

Scientists said the work has implications for understanding the structure of materials made up of random fibers, such as wool and fur and will have resonance with the computer graphics and animation industry, where the representation of hair has been a challenging problem.

In separate research, Stanford mathematician Joseph B Keller, studied why jogger´s ponytails sway from side to side, instead of up and down when they run. He found that the up-and-down motion was too 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 movement to become side-to-side swaying.

Goldstein got the idea for his new research after being contacted by Unilever, the multinational corporation whose products include soaps and shampoos, asking him about collaboration on research on the basic properties of hair.

“Somehow, 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,” remarked Goldstein.


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