With the help of mathematics: what is the least painful way to detangle hair

WASHINGTON.- Lakshminarayanan Mahavedan is a true genius of the Harvard University who has studied things like the shape of an apple, the convolutions of the brain, and why cereal flakes stick to a bowl of milk.

Now moms and dads will be happy to know that Mahadevan and his team of researchers have also discovered a painless technique for brushing your children’s tangled hair – and your own.

The task arose from a personal challenge. Two decades ago, when he was working in the Department of Applied Mathematics and Theoretical Physics at Cambridge University, Mahadevan’s domestic duties included brushing the unruly hair of his five-year-old daughter.

And at one point, his daughter fired him. “It was a very tangled bush,” says the professor, who has little hair in all of this. “I didn’t do my homework very well, because I’m so impatient.”

That failure haunted him for 20 years. Finally, Mahadevan put together a team that applied mathematical models, performed experiments and recently published their results in the scientific journal Soft Matter.

Hair brushing is an ancient practice, perhaps more than 50 million years old. According to several scientific studies, the mammals of that time developed tiny claws to groom themselves, precursors of nails, to remove lice and remove ticks that were staying in their fur. About 5,000 years ago, the Egyptians already considered hair care as an important part of grooming, to the point of including numerous combs and brushes in the tombs to accompany the deceased to the afterlife. According to estimates, the hairbrush industry moves hundreds of millions of dollars annually.

“At least half of humanity combs their hair every day, and yet hardly anyone thinks about it,” says Professor Mahadevan, winner of a MacArthur Fellowship, also known as “the genius grant.” Among other things, the scientist is an expert in applied mathematics, physics, and organismic and evolutionary biology.

Hairdressers are in the front line of the fight against hair tangles. “We see it all the time,” says Francis Peña, who owns a men’s barbershop in Marblehead, Massachusetts. “Most guys don’t take too much care of their hair, they don’t comb and wash it as often as they should, so I end up untangling it myself.”

Michael Conti lives in Westminster, Massachusetts, and having three sons, he never had to think too much about combing or brushing long hair. But then a girl arrived, who is now six years old, and that’s when things really got “tricky.” Conti says she starts at the top of her head, “and I just try to rake down, but all I do is throw her head back and my daughter groans and wonders what I’m doing.”

Conti is a mechanical engineer and tries to go through the comb slowly, knot by knot, centimeter by centimeter. The technique works, he says, “but it takes forever.” And his wife Sarah, a web designer, isn’t doing much better either: “My daughter just yells at me,” says her mother.

Their daughter Gianna does nothing to disguise her parents’ hair-brushing performance. “Terrible,” she says. She finally asked for a haircut short enough to spare everyone that headache. “So now I do my own hair.”

The enigma of the tangled hair came to Professor Mahadevan’s laboratory three years ago, while he was pondering the way birds build their nests. That research necessarily led to the question of entanglements, which also occur at the microscopic level in DNA helices and in the electromagnetic flux lines that crisscross the cosmos.

“The natural question is the following: how do you describe an entanglement?”, says Professor Mahadevan. “How do you dynamically modify an entanglement, and in this very specific context, basically how do we untangle it?”

To understand the physics of the problem, Mahadevan’s team simplified the challenge to two relatively wavy strands of hair, twisting around each other like the double helix of DNA.

They passed a single tooth of a comb through the strands. As the comb moved down and the tangle worsened, so did the tension in the scalp. Scientists calculated something called “link density”: the higher the bond density, the worse the tangle and the more likely the child in question will end up crying during combing.

The team’s finding, as published, is aimed at a much smaller audience. “The decoupling of the homochiral helices during this process can be quantified in terms of the Calugareanu-Fuller-White (CFW) theorem which states that Lk=Tw+Wr, where Link (Lk) quantifies the number of oriented crossings of the averaged strands over all projection directions” and so on.

A few blocks from Harvard, in the Massachusetts Institute of Technology (MIT), Daniela Rus, another MacArthur Grant recipient, noted those calculations. She is a professor of electrical and computer engineering, and director of the Computer Science and Artificial Intelligence Laboratory at the prestigious MIT.

His work points more towards automation, that is, to ensure that the task is no longer in the hands of parents. Conclusion: the researchers are building a “hairdressing robot.”

Rus’s lab is full of graduate students hunched over computer screens. On the work tables there are mechanical arms. One of the robotic arms is attached to a brush and runs through five different wigs — straight, wavy and curly — placed on a mannequin.

That lab brush is equipped with sensors that detect the tension. That tension is read as “pain” and is used to determine whether the mechanical arm should advance with more or less force. The machine does a great job, says Rus. “It’s slower and more rhythmic than when we do it ourselves.”

Science has finally discovered what most parents are learning through trial and error. “Start at the root of the hair, with short strokes,” says Rus. “And as we go along, those pulls get longer.”

The entanglement of DNA, the universe and human hair has yet to be fully unraveled. “We’re not that far yet,” says Professor Mahadevan. “But we or someone else will pick up the end of that ball.”

By Douglas Belkin

(Translation by Jaime Arrambide)

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