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March-April 2008
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< previous | 1 | 2 | 3 | 4 Mahadevan, who grew up in India, tells a traditional story about Krishna where mud becomes metaphor. “In Hindu mythology, Krishna is divine,” he begins. “However, because there was a prophecy that he would overthrow an evil king, his origins when he was a baby were hidden from almost everybody. So when Krishna was born, his mother surreptitiously sent him away to be brought up by a foster mother who didn’t know who he was. As in all mythologies, there were premonitions [of greatness], but growing up with his foster mother, he would go out like all children and play in the mud. One day he started to eat the mud, putting it in his mouth. And his [foster] mother, from afar, said, ‘Don’t do it.’” Krishna kept eating the mud. “Again [she] said, ‘Don’t do it,’ and yet he continued. So she came up to him, and when she opened his mouth to take out the mud, she looked—and she saw the universe.
Photographs by Jim Harrison Using an origami model, Mahadevan demonstrates how folds in the leaves of a hornbeam or a beech are coupled, allowing them to easily open and close. “Without ever claiming all the grandeur that the story actually suggests,” says Mahadevan, “you just have to look and you will find interesting things everywhere.” Mahadevan has a particular knack for choosing interesting problems—and for coming up with solutions that are “as simple as possible, but no simpler,” says his doctoral student Madhav Mani, echoing Einstein. “When he looks at a problem, he is able to strip away all the junk and say, ‘This is the essence of the phenomenon we are looking at.’” It is a gift Mahadevan shares with students, from first-years in his freshman seminar, “Science of Everyday Life,” to graduate students in Engineering Sciences 216, “Biological Dynamics,” to his half-dozen most advanced post-doctoral fellows. “[Applied mathematicians] are all technically as good as each other,” says Mani, “but the ability to see, for example, that there is a competition between two areas of physics leading into one problem,” either one of which might offer an answer, and to focus on the one that holds the key to the solution: “this is the kind of thing that you want to learn as a Ph.D. student.”
In his office, Mahadevan keeps an origami model of a leaf made of green paper, patterned like the leaves of beech or a hornbeam. “This is a very clever design,” he says admiringly. “You can open and close the leaf by playing with just one place.” Mahadevan manipulates the model with his fingers. A plant would use water pressure to do the same thing. “The folds in this leaf” he says, “are very different from those in a map, in which each fold is decoupled,” meaning that every fold can go either way—which is what makes road maps hard to close. In the leaf, on the other hand, all the folds are coupled to each other, so they open and close in unison. This same pattern, known as Miura-ori (after the Japanese engineer Koryo Miura), appears again and again in nature—in flowers and insect wings, for example, which have nothing in common at the genetic or molecular level—an apparent case of convergent design. Mahadevan thinks he knows why. Looking at a “serendipitous experiment” associated with the wrinkling of gelatin, he found that this Miura-ori pattern arises naturally as the most efficient solution to a physical problem. When gelatin shrinks in two directions, the pattern develops spontaneously. “The analogue in biology would be that cells are shrinking, or expanding or reproducing, or dying” at a greater rate on a surface layer of a tissue than on an inner layer, forcing the cells to push together. “That is all you need,” says Mahadevan. “You don’t need to control where this fold or that fold will occur. The rest just follows, following the laws of physics, nothing more complicated than that”—a plausible explanation for why these patterns appear in so many places. The underlying principle, which involves the differential expansion or contraction of cells, can also produce mechanical effects. “If you wet a pine cone, it will close,” Mahadevan points out. “If you dry it, it will open.” This occurs when water swells one part of the pine cone more than another, as in a bimetal thermostat where two strips, made of different metals that expand at different rates in response to temperature, have been stuck together. When the metal on one side swells, the thermostat bends away to the opposite side to accommodate the expansion. When the metal contracts, the apparatus swings back the other way. A pine cone with seeds generally stays closed when it is on the tree, but as soon as it falls, it dries and opens, and the seeds start to disperse. “These are solutions to problems in biology, from which we may learn something once in while.”
The office also includes a collection of toys and interesting objects that once amused his children on weekends when they accompanied him to work. There is a purple cape, purloined from his daughter, that drapes and folds dramatically; a model of a biological structure that has kinks in it and can rotate; tubes for making mathematical models; and interlocked metal rings that form a toroidal bubble—one shaped like a donut—when they spin around a piece of flexible tubing that loops through the donut’s center hole. Frequently, the objects illustrate ideas he has been working on. “This is a crochet made by a mathematician at Cornell, Daina Taimina,” he says, holding up an object that resembles a bell with a wavy lower edge, or more fancifully, a flamenco dancer’s frilled dress. He uses it to illustrate a fundamental question in geometry, but also to think about the folding of the brain during embryogenesis, when gray matter packs itself into the tight confines of a skull. 
Photograph by Jim Harrison A purple cloak demonstrates— dramatically—how fabric drapes and folds. Crochet, he explains, is made by knotting thread or yarn. “Suppose I wanted this object to be a flat, circular disk,” he says. “Starting at the center, I would increase the number of knots linearly with the radius,” to make, in effect a round hot pad. If, instead, as he worked out from the center, he decreased the number of knots at the perimeter, a spherical cap would start to form. But when the number of knots at the perimeter increases exponentially, you begin to get something like Daina Taimina’s creation. The exponential growth at the edges leads to excess material that can be accommodated only by having folds. “This is a very deep mathematical result that goes back, in fact, to David Hilbert and Henri Poincaré,” Mahadevan explains. He is interested in what happens next. “Geometers and mathematicians will tell you that as the structure grows more and more it will self-intersect, because there is too much material, and the only way to accommodate it all is basically to have it go through itself. Of course, this cannot happen with real material. Instead, the sheet forms complex folds.” In the case of the brain, for example, “you have a flat sheet that is growing rapidly” inside an enclosed structure, the skull. “Brain folding is a difficult problem, which a number of research groups are working on,” Mahadevan says. Rather than tackle it directly, he has been climbing a smaller hill: he has been studying the growth of seaweed. Kelp, the blades of which can grow as long as 10 meters, lives in shallow ocean waters. Typically, these algal blades are flat in the center and ruffled at the edges. But the amount of ruffling depends on the environment, says Mahadevan, whose interest in the problem was sparked by University of California, Berkeley biologist Mimi Koehl while he was on sabbatical there. If moved into fast-flowing water, the plant loses its ruffles. Transplant the same algal blade back to areas with gentle currents, and the ruffles regrow. Researchers think that the advantage to the plant is to ensure that it wiggles—exposing as much of its surface as possible to the sun, maximizing photosynthetic activity—but doesn’t flap so much that it breaks in strong currents. “Of course the plant is not a sentient being that knows when to grow or not,” Mahadevan explains. “The real question is, how does stress connect to growth?”
Structure is a common theme in Mahadevan’s work: how shape arises and how things change shape and flow. “How does structure essentially force certain things to happen,” he asks, “predetermining or very strongly constraining function in physics, chemistry, engineering, mathematics, geometry, biology, and medicine?” Thus Mahadevan’s interest in the way plants move water has led to one collaboration exploring how they grow pollen tubes and other structures using hydrostatic pressure, and another exploring how the design of veins in a plant’s leaves work to control evaporation in different climates. Working in his lab with John Higgins, a clinical fellow in pathology from HMS, and with his experimentalist colleague Sangeeta Bhatia, who is at MIT, he is studying the physical basis for sickle cell disease—in which hemoglobin sometimes forms a flexible rod inside the balloon-like red blood cells, causing cells to take on the characteristic sickle shape that can block capillaries—using a small device that mimics symptoms of the disease. The goal is to understand the mechanisms associated with flow blockage, while also developing technologies that might be useful for individualized medical testing and for testing potential drugs without using human subjects. And based on experiments in the HMS laboratory of Sabbagh professor of systems biology Timothy Mitchison, he has come up with a new way of thinking about the inside of a cell. “Is it a fluid? Is it a solid? My theory, which has directly testable predictions, is that it is neither, but rather a fluid-infiltrated sponge.” Mahadevan knows that such wide-ranging interests leave him exposed to the charge of dilettantism. To which he replies, “I hope that I am not a butterfly, but more like a bee. A bee pollinates. I move from flower to flower. I hope, once in a while, I am successful.” The kelp is an example of a recurrent theme among Mahadevan’s interests: how growth leads to form, and how form changes function in the context—in this case—of having more or less light. He has developed a mathematical model of cellular growth in an algal blade, which shows that the pattern of ruffling arises naturally when the rims of a blade grow faster than the interior. The rims form a series of saddles, similar to the waves in the crochet, in order to accommodate their greater length. “The same thing happens when I take sheets of paper and wet them and dry them and different parts wet and dry differently,” Mahadevan says. “Spray water on a paperback in the bathroom and when it dries it is very difficult to keep it shut again…. The point is that there is a unity to these ideas. Science is a continuum, and the goal should be to explain the largest number of observations with the smallest number of assumptions.” Mahadevan moves easily from modeling the way kelp grows to a simple understanding of how it sways in a current, based on his work examining how flags flutter. He holds a thin sheet of paper to his lips, and blows softly. The paper gently rises. Then he blows rapidly, and the sheet begins to flutter wildly. “There are many problems where you move from a steady behavior to an oscillation,” he observes. In the 1930s, as planes began to fly faster and wings sometime broke off as a result of high-speed vibrations, aeronautics engineers devised ways to design around the problem. Similar wind-driven oscillations famously brought down the Tacoma Narrows Bridge in 1940. Mahadevan and a postdoctoral fellow, Médéric Argentina, now an assistant professor in France, realized the flutter of a flag could be used to unify a whole class of problems in a single conceptual idea: that a fluid such as air or water can actually couple different modes of energy. “What a flag does,” says Mahadevan, “is essentially convert a uniform DC [direct current] flow to AC [alternating current], if you think of it in terms of electrical analogies.” The steady flow of the wind is converted into the oscillations of the flag. The simplest way to think of this is to imagine that the flag is rigid. Mahadevan raises his arm to demonstrate, using his elbow as a pivot point. A wind blowing would push his arm first up, then down, as it overshot a position parallel to the direction of the wind. In this way, a flow can be converted to an oscillation, “something that has been understood since the time of Lord Kelvin.” But of course a flag is not rigid. “Even in the absence of a fluid, if you hit the flag it will deform and then come back to rest.” The combination of this elastic response and the fluid-driven response led to “a precise description of how you can push energy from the fluid into the solid and make it vibrate,” he says. And of course one can turn the problem around, turning AC movements into DC locomotion. “Think of an eel that [propels itself forward] using waves of undulation that propagate along its body.” In a recent paper with his student Zengcai Guo, Mahadevan described how snakes “swim” this way on land, by “pushing off against protuberances.” And more whimsically, motivated by watching skates and rays skimming along the bottom of an aquarium, he has calculated, with student Jan Skotheim and Argentina, the conditions under which undulating carpets—actually thin films—could be made to fly. Mahadevan is a testament to science driven primarily by curiosity. “Nobody questions whether art or music is useful and important and relevant,” he says. In the same way, he believes, there ought to be room for science pursued simply because it is part of human culture. As a teacher, he aspires to whet and satisfy the natural scientific curiosity that drives such inquiry. No degree in mathematics is required. “To be able to experience some of the wonder and the mystery,” he says, “you just have to probe the immediate environment around you.”
Jonathan Shaw ’89 is managing editor of this magazine. Readers interested in additional information on L. Mahadevan’s work may wish to visit his website, www.seas.harvard.edu/softmat.  
 
 
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