Written by NaturallyCurly Co-Founder Michelle Breyer for her blog, The Curly Connection

###### PHOTO: MIT RESEARCHERS JAMES MILLER AND PEDRO REIS

Growing up, I always envied the silky-haired girls with ponytails that swished. Darcy Waterman, a girl on my high school track team, had a long sleak sun-streaked ponytail that cascaded down her back, swinging from side to side as she jogged effortlessly in front of me. That ponytail mocked me. My puff of a ponytail was immobile – like a bonsai tree on top of my head.

Little did I know that the difference in our two ponytails was something being studied by scientists around the world who were trying to tease out the physics of curly hair.

Curly girls know first hand how challenging their coils and ringlets can be. The complexities of curls, coils and waves also challenged the film and computer animators who tried to recreate them.

Most of the heroes and villains in animated films had hair that was extremely rigid straight hair, swinging to and fro. It was rare to see an animated character with bouncy, curly hair, since computer animators didn’t have a simple mathematical means for describing it.S o it was was big news when scientists last month announced they had created the first detailed model of a 3D strand of curly hair was recently created – something  that had vexed film animators for decades.

Researchers at MIT, in Cambridge, Mass., and the Université Pierre-et-Marie-Curie (UPMC) built their model using flexible rods to examine varying degrees of curliness.

“Our work doesn’t deal with the collisions of all the hairs on a head, which is a very important effect for animators to control a hairstyle,” study co-author Pedro Reis, an assistant professor in MIT’s department of civil and environmental engineering, said in a statement. “But it characterizes all the different degrees of curliness of a hair and describes mathematically how the properties of the curl change along the arc length of a hair.”

Using lab experimentation, computer simulation, and theory, the team identified the main parameters for curly hair and simplified them into two dimensionless parameters for curvature (relating to the ratio of curvature and length) and weight (relating to the ratio of weight and stiffness). Given curvature, length, weight, and stiffness, their model will predict the shape of a hair, steel pipe, or Internet cable suspended under its own weight.

As a strand of hair curls up from the bottom, its 2-D hook grows larger until it reaches a point where it becomes unstable under its own weight and falls out of plane to become a 3-D helix. Reis and co-authors describe the 3-D curl as a localized helix, where only a portion of the strand is curled, or a global helix, if the curliness extends the entire length up to the head.

A curl can change phase — from 2-D to 3-D local helix to 3-D global helix, and back again — if its parameters change. Because a strand of hair is weighted from the bottom by gravity, the top of the strand has more weight under it than the tip, which has none. Thus, if the weight on a hair is too great for its innate curliness, the curl will fail and become either straight or helical, depending on the strand’s length and stiffness.

For the curvature study, Miller created flexible, thin rods using molds as small as a bottle of Tabasco sauce and as large as the columns in MIT’s Lobby 7 (about a meter in diameter). He injected a rubber-like material inside hollow flexible tubing wrapped around these molds. Once the rubber material cured and the tubing was cut away, Miller and Reis had flexible polyvinyl thin rods whose natural curvature was based on the size of the object around which they had been wrapped.

“One of the key issues was how to handle the distribution of intrinsic curliness found in real hair,” reis said.

They were able to focus in detail on the properties of a single curly hair under gravity.

“The fact that I am bald and worked on this problem for several years became a nice running joke in our lab,” Reis says. “But joking aside, for me the importance of the work is being able to take the intrinsic natural curvature of rods into account for this class of problems, which can dramatically affect their mechanical behavior. Curvature can delay undesirable instability that happens at higher loads or torsion, and this is an effect that engineers need to be able to understand and predict.”

In addition to helping animators create more realistic curls, this technology also also could be used by engineers to predict the curve that long steel pipes, tubing, and cable develop after being coiled around a spool for transport. In the field, these materials often act like a stubborn garden hose whose intrinsic curves make it behave in unpredictable ways. In engineering terminology, these items — and hair — are all examples of a slender, flexible rod.

This work was funded by the National Science Foundation, Schlumberger, the MIT-France Program, and a Battelle-MIT postdoctoral fellowship.