Math untangles the mystery of ponytails

U. WARWICK (UK) — The Ponytail Shape Equation explains how hair behaves in a bundle, and may have applications in textiles and animation.

University of Warwick and University of Cambridge physicists have quantified the curliness of human hair and developed a mathematical theory that explains the shape of a ponytail. The research, published in the journal Physical Review Letters, provides the first quantitative understanding of the distribution of hairs in a ponytail.


To derive the Ponytail Shape Equation, the scientists took account of the stiffness of the hairs, the effects of gravity, and the presence of the random curliness or waviness that is ubiquitous in human hair

Together with a new quantity described in the article—the Rapunzel Number—the equation can, they say, be used to predict the shape of any ponytail.

The research by Raymond Goldstein from the University of Cambridge, Robin Ball from the University of Warwick, and colleagues, provides new understanding of how a bundle is swelled by the outward pressure that arises from collisions between the component hairs.

This has important implications for understanding the structure of many materials made up of random fibers, such as wool and fur. The research will also have resonance with the computer graphics and animation industry, where the representation of hair has been a challenging problem.

“It’s a remarkably simple equation,” explains Goldstein, a professor in Cambridge’s department of applied mathematics and theoretical physics.

“Our findings extend some central paradigms in statistical physics and show how they can be used to solve a problem that has puzzled scientists and artists ever since Leonardo da Vinci remarked on the fluid-like streamlines of hair in his notebooks 500 years ago.”

“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.”

“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,” adds Goldstein.

“We now have the first quantitative description of this phenomenon and how it competes with gravity.”

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