TRICKY TANGLE: A mathematical model shows that straight hair tangles more often than curly hair.
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On a cool Saturday afternoon at the überhot Garren hair salon in New York City a few masters of fashion were debating something many would call obvious: Which is more likely to tangle—curly hair or straight hair?

The seemingly straight answer, if you will: "When we put a [model] in a wind machine, we can still put a comb through her hair—if it's straight. Curly? Forget it," says Robert Vasquez, a hairstylist who specializes in what his industry terms "difficult" hair.

But what appears, at first thought, to be obvious can become surprisingly fuzzy. "I'm not convinced that curly hair tangles more," says Steven Fernandes, who combs up to 20 manes daily at Garren. "Straight hair is lighter and strands move individually, but curly hair moves as one movement and so is less tangled."

Jean-Baptiste Masson, a young brain imaging researcher at the École Polytechnique in France, recently pulled apart this question as a class exercise in mathematical modeling.

With hundreds of thousands of micro-thin strands colliding in every direction, hair is an unusually complicated system—a set of independent objects working as an integrated whole. And, unlike other  systems such as fluids and solids, hair mechanics remains unsolved, with no widely accepted model to explain it. "I needed a problem for my students, and I thought hair is something that could be simplified," Masson says.

He enlisted two hairdressers to count the number of tangles on 212 heads over three weeks. The stylists were instructed to look for true tangles, namely a clump that resisted the draw of a comb but was not a cluster of hair—like a ringlet, or curl.

Based on these criteria, curly hair averages about three tangles per head whereas straight averages more than five tangles. The surprising results, and a mathematical model of tangling, were recently published in the American Journal of Physics.

In Masson's model, it turns out that even though curly strands meet more often than straight strands, the angle at which two straight hairs meet is the angle most likely to lead to tangling.

But there's no particular explanation for why different angles lead to different tangling patterns, argues Alain Goriely, professor of mathematics at the University of Arizona in Tucson, who studies the math of biological systems.

Masson explains that the optimal tangle angle is one that is large enough to hook the microscopic, fishlike "scales" that coat hair cuticles. If this angle is too narrow—meaning the two hairs are nearly parallel—they won't lock.

It turns out that Masson's model predicts the real-world hairdresser data with surprising accuracy.

Masson's "math lesson" has inspired computer scientists like Florence Bertails, an expert in hair behavior at The French National Institute for Research in Computer Science and Control, to consider using models like the one Masson devised to inform her more complicated computer algorithms.

Back at Garren, Fernandes has the last point, "You know what the real issue is? The real issue is fine versus coarse. Curly or straight, fine hair is what tangles. The cuticle is open and puffy, like Velcro—it'll stick to anything."

The others nod their heads dramatically. "Oh yes, that's it," says Vasquez, "dry, fine, chemically treated hair tangles the most."