The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse

by Jennifer Ouellette

  Their epic battle inspired a local poet and wag named Josep Carner to compose a rhyme describing a fictional “Mrs. Comes,” who has been given a grand piano for her newly decorated home. Not only is its sheer size problematic, it has “no style” and disrupts the harmony of the space. Mrs. Comes asks the great Gaudi for a solution. He advises her to play the violin. Satire it may be, but the tall tale captures the essence of Gaudi: He expected others to adapt to his artistic vision, not the other way around.

  It was while designing the Church of Colonia Güell on the outskirts of Barcelona that Gaudi developed his unique method for determining the best curvature for his many arches and ribs in the church’s crypt, taking his inspiration from gravity. He devised a “hanging model” approach to calculate the loads on the arches. It was an elaborate system of interconnected threads, representing the columns, arches, walls, and vaults of any given design, from which he suspended sachets filled with lead shot to mimic the weight of various building components. Not surprisingly, the end result was often a catenary. Catenary shells are still used in structural engineering today.

  Gaudi’s method wasn’t sufficient to help him design the more complex, double-curvature vaults used in the nave of his unfinished masterpiece, La Sagrada Familia, a massive cathedral still under construction 125 years later. (In fairness, 180 years passed before the famed Notre Dame cathedral in Paris was completed.) It took Gaudi 10 years to complete his design, reworking his blueprints over and over until he was satisfied with the result. His plans called for eighteen towers (twelve for the apostles, four for the evangelists, one for Mary, and one for Jesus.) Each tower features intricate geometric designs and small ornamental sculptures, and the buttresses inside the nave look for all the world like tree trunks.

  All told, Gaudi worked on the cathedral for forty-three years, twelve of them devoted exclusively to the project. For the last years of his life, he actually lived in the structure’s crypt. Poor Gaudi met with an ignominious end: He was run over by a tram on June 7, 1926, while walking to the construction site, and he was so raggedly dressed that nobody recognized the famous architect. (Several taxi drivers refused to drive such a vagabond to the hospital, and were later fined by municipal police for their refusal to assist the injured man.) He wound up at a pauper’s hospital, and although his friends tracked him down a day later and tried to move him to a better facility, Gaudi refused, declaring, “I belong here among the poor.” He died three days later and was buried within the Sagrada Familia.

  No doubt Gaudi would be gratified to learn that his masterpiece is nearing completion. Jordi Bonet, director of La Sagrada Familia since 1985, has said the interior will be completed in 2010, with plans to mark the occasion with the celebration of Mass in the main nave. After that, only one last tower must be built: the 550-foot-tall Tower of Jesus, slated for completion in 2026.51 At least part of the delay was due to the fact that contractors initially couldn’t figure out how to physically build some of the bizarre structures Gaudi designed on paper. And not only did Gaudi invent his own system for calculating his catenary shapes, he did those calculations without the benefit of modern computers.

  WALKING ON EGGSHELLS

  John Ochsendorf remembers the first time he stood on top of the domed vault of the chapel at King’s College, Cambridge. “You’re standing eighty feet off the ground on a thin piece of stone,” he recalls. “You can even feel small vibrations. And you can’t help thinking, ‘The nerve of these people!’ ”

  Ochsendorf is a structural engineer at the Massachusetts Institute of Technology and a historian of architecture and construction. “These people” are the long-dead members of England’s masonry guild who built the chapel roof around 1510. It’s not difficult to see why he is so impressed with their engineering skills: The chapel’s roof spans nearly 15 meters, yet it is only 10 centimeters thick—similar to an eggshell in terms of its radius to thickness ratio. “These [early arch builders] developed a very real science of construction to attain a high degree of stability,” says Ochsendorf. “I’m simply in awe of the fact that we haven’t surpassed it yet.”

  Modern architects have devised their own tricks of the trade. Like Gaudi before him, contemporary Swiss architect Heinz Isler creates what he calls reversed “hanging membranes” to design the delicate, thin-shelled dome structures for which he is justly famous. After pouring liquid plastic onto a cloth resting on a flat, solid surface, he lowers the surface, leaving the plastic-covered cloth to hang in pure tension, suspended from its corners. The plastic hardens, freezing that position. Once it has dried, Isler turns the solid shell model upside down, and that form becomes the basis for his design—a form of experimental calculus.

  Ochsendorf’ s work is aimed at adapting a popular computer graphics tool to help unlock the elusive secrets behind the arches and domes of Gothic cathedrals. Along with his then-graduate student, Axel Kilian, Ochsendorf adapted a technique called particle-spring modeling, in which virtual “masses” at the various “nodes” of a design are connected by virtual “springs.” These bounce around until they find equilibrium and are able to support the requisite loads, just like Gaudi’s hanging chain. CGI animation already uses such particle-spring models to re-create the movement of fabrics and hair, because animators need to map out how forces flow in different directions in real-time 3D, and in an interactive format. Remember the scene in Star Wars: Episode III—Revenge of the Sith where Yoda fights an adversary while wrapped in a cloak? The movement of Yoda’s cloak was designed using a particle-spring model.

  Ochsendorf and Kilian realized there were parallels between the fabrics that CGI animators model and Isler’s hanging membranes. A length of cloth is strong under tension, but if you push on it (compression), it simply crumples. What Ochsendorf needed was something with precisely opposite properties, so he worked out a way to turn the fabric model around. This allowed him to model architectural structures, specifically Gothic cathedrals.

  He’s already scored some successes with his prototype program. Ochsendorf used it to demonstrate that the domes of the Pines Calyx conference center near Dover, England, would stay in compression under all possible loadings, thereby satisfying stringent safety regulations. Open for business since 2008, the center is topped by domes made from clay tiles glued together edge to edge. Those domes span 15 meters, yet the tiles are only 15 centimeters thick and required no supporting framework during construction. “Without Ochsendorf’ s program these remarkable thin-shelled shallow domes would not have been allowed to be built,” Alistair Gould told me. Gould is a member of Helionix Designs, a firm based in Kent that designed the building.

  Eventually Ochsendorf hopes to provide designers with a technique that could lead to revolutionary architectural designs and more environmentally friendly buildings. Many modern buildings have a severe impact on the environment. Steel corrodes with time, and the manufacture of concrete produces quantities of greenhouse gases. Ochsendorf ’s software program has already demonstrated that certain buildings could have been built with much less material. In essence, the program finds the solution to an optimization problem for the materials.

  Take MIT’s Kresge Auditorium, designed by Saarinen in 1955. It has a domed roof made of concrete 15 centimeters thick. After analyzing the geometry of the dome and feeding the measurements into his hanging-chain model, Ochsendorf reckoned that it could have been built with half the thickness of concrete, resulting in significant savings in building costs and reduced environmental impact—without sacrificing the artistry. He made similar findings about MIT’s new computer science building, designed by Frank Gehry. The building features columns leaning in every direction, and the structure used roughly 30 percent more material than would have been needed if his program had been used to find where the lines of force naturally fall, Ochsendorf insists.

  THE QUEEN’S GAMBIT

  One of the most famous optimization problems can be found in Virgil’s Aeneid. A Phoenician woman named Dido was forced
to flee her homeland after a tyrannical brother murdered her rich husband and tried to seize her wealth. Dido didn’t exactly travel light: She “fled” via several boats filled with her belongings (including her late husband’s stash of gold) and numerous attendants, eventually landing on the coast of Africa, where she hoped to start a new life. Her reception by the natives was frosty at first—perhaps they’d encountered would-be colonists before—but she struck a shrewd bargain with their king, offering a substantial sum of money in return for as much land as she could mark out with the hide of an ox upon which to build her own city.

  It may be that the king bought into the stereotype that women aren’t inherently good at math; figuring he was getting the best of the deal, he agreed. Dido promptly took her oxhide and cut it into thin strips, which she joined together into one long strip. Using the seashore as one edge for her promised tract of land, she then laid the skin into a semicircle, thereby ensuring that said tract was significantly larger than the African king had thought possible. And on that site she founded the great city of Carthage (near modern-day Tunis), reigning as its queen. In mathematics, this is known as the isoperimetric problem: How does one enclose the maximum area within a fixed boundary?

  Ah, but just how do we know that Dido’s semicircle did indeed enclose the largest area given the length of that long thin strip? Calculus, of course—specifically, we must solve a maximization problem using the calculus of variations. Let’s start with a simpler idealization to demonstrate the basic method. We’ll assume that Dido’s strip of oxhide is 600 feet long, and she wants to enclose the largest possible rectangular area in which the seashore provides a boundary along one side. Even in this simplified version, there are many different possible permutations she could make with that 600 feet of oxhide: long and narrow, tall and thin, and everything in between. What shape is the likely candidate for giving her the optimal square footage?

  We have to start somewhere, so let’s take the shape of a square as a point of reference. By definition, this means that Dido would need 200 feet of oxhide for each of the three sides, with the shoreline of the Mediterranean Sea serving as the fourth side. That gives us an area of 40,000 square feet, as area depends on length and width. However, we have no way of knowing (yet) whether this is indeed the optimal shape. So we begin varying the shape ever so slightly in different directions. For instance, if Dido arranges her oxhide to measure 201 feet down the width of two opposite sides and 198 feet across the length, she would have an area of 39,798 square feet—slightly less than a perfect square. Dido decides to test this further, and adjusts her dimensions in a different way. This time, the two opposite sides measure 199 feet in width, and the third side measures 202 feet across. The answer: 40,198 square feet. Clearly the square will not give her the most possible area.

  The crucial point is that the question posed has to do with change in area, not simply the static values of the area—that way lies madness, for we would be randomly computing areas for different configurations in hopes of stumbling on exactly the right one. It is far more useful to consider all possible areas created by all possible configurations (i.e., an infinite number). We have now seen countless times how the derivative applies to any case where a change in one quantity produces a corresponding change in another quantity; the derivative measures that rate of change.

  Dido can reduce the problem to a simple function: Knowing she has 600 feet of oxhide, once she chooses a width for her enclosure (w), the remaining oxhide will be evenly divided to make up the length of the plot of land. How do we translate this into an equation? We know that area equals width multiplied by length. So given the variables for width (w) and length (L) of the configuration, as well as the total amount of oxhide (600), we come up with the function 600w − 2w 2. This is the function she would use to determine what the area would be for any given configuration she chose. Graph that function by plugging in various values for w—ranging from a width of 0, to a width of 300—and you get a pretty curve (the “face” of Dido’s function). This greatly narrows the possible solutions.

  Now all we need to do is find a spot along that graph where the rate of change is 0. Recall that the slope of the tangent line along a curve is equivalent to the derivative. The place where the derivative is 0 will therefore be at the very top of the curve, where the tangent line is a horizontal line and hence has no slope. And that point occurs where w = 150 feet. Ergo, Dido’s plot of land should be 150 feet wide to get the maximum possible rectangular area (45,000 square feet) on which to build the city of Carthage.

  This saves Dido the trouble of having to calculate the areas of all possible rectangular shapes. She simply graphs the function and then looks to see which values give a slope (derivative) of 0. Even if there are four such spots, instead of one in this particular case, that narrows the possibilities considerably. She can certainly calculate the areas of four possible shapes to determine the best possible width for her planned city.

  But remember that the optimal shape is not a rectangle, but a semicircle. To find the true optimal shape, Dido must use the calculus of variations. Just as with the brachistochrone problem, it is necessary to integrate over all possible curves—not just rectangles—to pluck the correct answer from among the infinite hordes. A semicircle with a length of 600 feet has a radius (distance from the circle’s center to its arc) of 191 feet. Since we know a full circle with this radius would have an area determined by πr2, a semicircle has an area determined by ½ πr2. So a semicircle yields an area of 57,296 square feet.

  Fans of Virgil know that things did not end well for Dido, queen of Carthage. She rejected the African king’s offer of marriage, only to foolishly fall in love with the wily Aeneas, who wound up in Carthage with his fellow surviving Trojans after the fall of Troy. But Aeneas abandons her to fulfill his manifest destiny of founding the Roman Empire. A heartbroken Dido builds a funeral pyre, curses Aeneas, and falls on a sword given to her by her fickle lover. Aeneas and his men watch the glow of her burning pyre from their departing ship, unaware of Dido’s suicide.

  Later in Virgil’s magnum opus, Aeneas travels to the underworld and runs into his former lover’s shade, but she refuses to acknowledge him, still bitter at his abandonment. The poet T. S. Eliot once called this “the most telling snub” in Western literature. I think it shows most clearly that hell literally hath no fury like a woman scorned—particularly a formidable woman like Dido, capable of outwitting an African king with an early conceptual harbinger of calculus, centuries before Newton and Leibniz invented it.

  9

  Surfin’ Safari

  I tried surf-bathing once, but made a failure of it. I got the board placed right, and at the right moment, too; but missed the connection myself. The board struck the shore in three-quarters of a second, without any cargo, and I struck the bottom about the same time, with a couple of barrels of water in me. None but natives ever master the art of surf-bathing thoroughly.

  —MARK TWAIN,

  Roughing It

  Few people are aware that novelist Samuel Clemens—better known by his nom de plume, Mark Twain—was a great admirer of the sport of surfing, or, as he called it in his travelogue Roughing It, surf-bathing. He even tried his own hand at surfing, with predictably dire results: He wiped out and swallowed a hefty helping of salt water for his trouble. I can empathize, as I come up sputtering from a spill for the umpteenth time during my own maiden stab at surfing in Kona, Hawaii. As I scramble back onto my beginners longboard, my self-appointed “surfing coach,” Milton Garces, casually rides a swell over and calls out a snippet of helpful advice: “You might want to move back a bit on your board; you were too far forward that time to ride out the wave!”

  He should know; waves of all kinds are his stock in trade, particularly sound waves and water waves. Garces is an acoustic oceanographer at the University of Hawaii-Manoa, specializing in the study of infrasound, aural signals at frequencies that lie below the range of human hearing (20 Hz to 22 kHz). Nature
has an entire palette of sounds that play constantly just beyond our ken. Human hearing is rather limited in range, but sound waves exist far beyond it. We can’t hear the ultrasonic shrieks of bats or the ultra-low-frequency waves of acoustic energy (infrasound) employed by elephants or tigers. Wind, water, earthquakes, avalanches, tornadoes, and hurricanes all produce infrasound, as well as audible sounds. To an acoustician, there’s no such thing as perfect silence.

  Most acousticians have a touch of the daredevil in them, almost by necessity: If you’re trying to study the propagation of sound waves, you’ve got to go where the waves are happening, even if that leads you to remote Mayan ruins or the foot of an active volcano. Garces is no exception. When he’s not exploding missiles at the White Sands Missile Range in New Mexico to better study the infrasonic waves that result from the explosion, he’s setting up infrasound sensor arrays around volcanoes in Ecuador, or on Japan’s Kyushu Island. Once he was caught napping in a Toyota Corolla in the vicinity of a volcanic eruption, resulting in some harrowing, ash-choked moments before he was able to drive to safety.

  So it’s not surprising that Garces is an avid surfer, along with just about everyone else in his Infrasound Laboratory (ISLA) on Hawaii’s Big Island. I have flown out to Kona to learn more about his lab, which is located right on the water’s edge, the better to collect data on incoming waves. While Maui is famous for its miles of sandy white beach, Kona’s shores are strewn with black lava rocks. The entire Big Island is the remnant of volcanic eruptions spanning thousands of years and is still home to active volcanoes. Locals like to place white shells against the black rocks to form pictures or spell out messages—Kona’s version of graffiti.