Ponytail Shapes Explained Through Math

A group of physicists at the University of Cambridge announce the development of the first mathematical model that can explain the shape of a ponytail, and can also quantify the curliness of human hair. Experts have been interested in the properties of hair since the time of Leonardo da Vinci.

According to the investigators, the new model could have a host of practical applications, primarily in the textile industry, the development of computer animation and the creation of new care products.

Details of the research appear in the February 13 issue of the esteemed scientific journal Physical Review Letters. What the model provides is a clear understanding of how human hair is distributed inside a ponytail. This set of calculations is called the Ponytail Shape Equation (PSE).

One of the most difficult things to account for was the random waviness (curliness) that is a trait of human hair. Experts then correlated this property with the effects of gravity and the stiffness of the hair itself. The latter is a measure that varies widely among individuals as well.

However, the equation also includes a new quantity, called the Rapunzel Number. With its addition, the PSE becomes capable of predicting the shape of any ponytail, investigators explain. The work was led by Cambridge professor Raymond Goldstein.

University of Warwick professor Robin Ball was a co-leader of the research. Together with their collaborators, Goldstein and Ball were able to create a simulation of how a hair bundle is swelled up by outward pressure produced when tightening a thick strand of hair together.

The new work may also provide a deeper understanding of how materials based on random collections of fibers behave. These include wool and fur, among many others. But the same type of data are used in computer animations, when creating a model of human hair and different fabrics.

“It’s a remarkably simple equation,” says Goldstein. He holds an appointment as the Schlumberger Professor of Complex Physical Systems at the Cambridge 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,” he concludes.