 <p class="lead" id="first">As anyone who has ever had to brush long hair knows, knots are a nightmare. But with enough experience, most learn the tricks of detangling with the least amount of pain -- start at the bottom, work your way up to the scalp with short, gentle brushes, and apply detangler when necessary.</p><p>L. Mahadevan, the Lola England de Valpine Professor of Applied Mathematics, of Organismic and Evolutionary Biology, and of Physics, learned the mechanics of combing years ago while brushing his young daughter's hair.</p><p><em>"</em>I recall that detangling spray seemed to work sometimes, but I still had to be careful to comb gently, by starting from the free ends," said Mahadevan. "But I was soon fired from the job as I was not very patient."</p><p>While Mahadevan lost his role as hairdresser, he was still a scientist and the topology, geometry and mechanics of detangling posed interesting mathematical questions that are relevant to a range of applications including textile manufacturing and chemical processes such as polymer processing.</p><p>In a new paper, published in the journal <em>Soft Matter,</em> Mahadevan and co-authors Thomas Plumb Reyes and Nicholas Charles, explore the mathematics of combing and explain why the brushing technique used by so many is the most effective method to detangle a bundle of fibers.</p><p>To simplify the problem, the researchers simulated two helically entwined filaments, rather than a whole head of hair.</p><p>"Using this minimal model, we study the detangling of the double helix via a single stiff tine that moves along it, leaving two untangled filaments in its wake," said Plumb-Reyes, a graduate student at SEAS. "We measured the forces and deformations associated with combing and then simulated it numerically."</p><p>"Short strokes that start at the free end and move towards the clamped end remove tangles by creating a flow of a mathematical quantity called the 'link density' that characterizes the amount that hair strands that are braided with each other, consistent with simulations of the process" said Nicholas Charles, a graduate student at SEAS.</p><p>The researchers also identified the optimal minimum length for each stroke -- any smaller and it would take forever to comb out all the tangles and any longer and it would be too painful.</p><p>The mathematical principles of brushing developed by Plumb-Reyes, Charles and Mahadevan were recently used by Professor Daniela Rus and her team at MIT to design algorithms for brushing hair by a robot.</p><p>Next, the team aims to study the mechanics of brushing curlier hair and how it responds to humidity and temperature, which may lead to a mathematical understanding of a fact every person with curly hair knows: never brush dry hair.</p><p>This research was supported by funds from the US National Science Foundation, and the Henri Seydoux Fund.</p>