The spectacular view is marred somewhat by our cramped conditions—the tram is a bit like a five-person ovoid coffin—and the disconcerting sensation of swaying whenever the wind picks up. The arch is designed to sway up to eighteen inches in the wind; it is closed to the public when the winds are particularly strong. For those unfamiliar with the principles of structural engineering, it may seem as if the arch were about to collapse. One woman in particular seems convinced of her imminent demise: Eyes shut tight, arms hugged tightly to her chest, she refuses her companion’s entreaty to take just one look before the tram returns to the ground. A bead of sweat is visible on her upper lip as yet another gust of wind hits the arch.
The poor woman need not have worried. Appearances can be deceiving. A well-constructed arch is actually quite stable. Leonardo da Vinci once observed, “An arch consists of two weaknesses which, leaning one against the other, make a strength.” The secret of how the arch stays up lies in its shape. There is a very specific geometric term for it: It’s essentially an inverted model of a flexible chain or rope suspended from two points. The noninverted curving shape is known as a catenary, derived from catena, the Latin word for “chain.” Leonardo’s codependent “leaning weaknesses” describe a delicate balance of opposing forces that gives rise to a surprising degree of structural stability.
Nature has its preferred shapes. Any chain suspended between two points will come to rest in a state of pure tension, much as surface tension causes a bubble to form a sphere. In a chain, tension is the only force between consecutive links, and that force inevitably acts parallel to the chain at every point along the curve—never at right angles to it (otherwise the chain would move). Inverting the catenary into an arch reverses it into a shape of pure compression. Masonry and concrete— standard building materials—break relatively easily under tension, but can withstand huge compressive forces. So the inverted catenary shape can be used to form structures like domes or arches that span a considerable horizontal distance.
Finnish architect Eero Saarinen didn’t copy the classic inverted catenary shape perfectly when he designed this St. Louis landmark; he elongated it, thinning it out a bit toward the top to produce what one encyclopedia entry describes as “a subtle soaring effect.48 But Saarinen’s catenary variation is more than an aesthetic choice. He designed it in consultation with an architectural engineer named Hannskarl Bandel, who knew that the slight elongation would transfer more of the structure’s weight downward, rather than outward at the base, giving the arch extra stability. He had good reason for doing so. An inverted catenary shape is extremely stable horizontally, but it is less so in the vertical direction—and the Gateway Arch rises a good 630 feet above its 630-foot base. A plaque near the site proudly declares that the Gateway Arch’s distinctive shape is described by this mathematical equation: y = 693.8597 - 68.72 cosh(0.010033x).
Any engineer can tell you that math is critical to building structures with the right size, shape, and balance of forces, and that geometry plays an important role in architecture. Yet equations for geometric figures weren’t even available until Fermat and Descartes devised analytic geometry in the seventeenth century, ensuring that millions of high school students would be required to take up compass and straightedge and learn about interior and exterior angle sum conjectures, along with the properties of trapezoids. Today it is a relatively simple matter to calculate the volume of the famous pyramid-shaped Luxor Hotel in Las Vegas, given the precise dimensions. Wikipedia tells me that the base of the pyramid is a 556-foot square and that the structure’s height is 350 feet. First we determine the area of the base by multiplying the width (556 feet) by the length (556 feet). Then we multiply that area by the height and divide that answer by 3 to get the total volume. No calculus required.
But how does one select the best possible design—one intended to optimize a particular feature, such as the optimal dimensions for a pyramid one could build given a certain amount of material, or the strongest possible shape for an arch—from a wide range of options? The tedious approach would be to painstakingly calculate each and every possible option, which would be incredibly time-consuming. With calculus, it’s possible to focus not on the absolute quantities one is interested in, but to look instead at how certain features are changing relative to each other—that is, to approach the problem dynamically. We can do this by determining the maximum and minimum values for the feature of interest to narrow the focus; the answer will lie somewhere in between.
ARCH RIVALS
Today we build almost exclusively with steel and reinforced concrete, and the design process is heavily reliant on mathematical modeling and engineering principles. In ancient times, arch builders employed a method of trial and error. Small stone arches were typically built around a curved wooden form. The builder would then lay stones or bricks around that wooden form, tracing the shape with pegs and string. Legend has it that whenever an arch was constructed in ancient Rome, the architect who designed it was forced to stand underneath as the wooden supports were removed, as a means of quality control. It was a terrific motivational tool: Design it right the first time, or the arch will fall and crush you. Builders of Gothic cathedrals had to figure out how to turn stones into stable structures held together only by the forces of compression, like a stack of children’s building blocks, and they did it without the benefit of analytical geometry or calculus. The oldest cathedrals have stood for a thousand years, so medieval masons clearly knew a thing or two about arch stability.
One assumes this practical knowledge was passed down through generations of builders. Yet the secret of the inverted catenary remained a mathematical mystery until the seventeenth-century English scientist Robert Hooke stumbled upon this solution to the question of an optimal shape for a stable arch. Hooke is best known for his skill at building microscopes and using them to examine the tiniest details of everyday objects, such as fleas. His exquisite drawings of what he saw through his microscopes appeared in his masterpiece, Micrographia. He also invented a reflecting telescope, the sextant, the wind gauge, and the wheel barometer, and he had a lifelong fascination with timepieces.
Despite these accomplishments, Hooke’s stature as a scientist was largely overshadowed by that of his contemporary rival, Isaac Newton. Their professional debates over the nature of light often became intensely personal: Hooke may even have tried to block Newton’s election to the Royal Society. Perhaps Hooke had cause to feel threatened: His more practical contributions to science were overlooked in favor of Newton’s mathematically oriented theories. Personal vanity may also have played a role: Newton cut a distinguished, imposing figure, while Hooke was small and hunched; even his friends described him in less than flattering terms.
Hooke’s pique at the lack of recognition by his peers might have been partially justified. In 2006, the long-lost handwritten minutes from meetings of the Royal Society between 1661 and 1682 were discovered wedged into a dusty nook in an old house in Hampshire, England. The manuscript laid to rest a long-standing controversy over whether Hooke or Christian Huygens had first designed a highly accurate watch with tiny spring mechanisms that eventually led to the first measurement of longitude. Hooke understood a great deal about the physics of springs, having devised the eponymous Hooke’s law:
Extension is proportional to force. So when Huygens claimed to have invented a spring watch in 1675, Hooke flew into a rage, claiming someone had leaked his earlier design to the Dutch scientist. The unearthed minutes include pages from a meeting on June 23, 1670, with a description of Hooke’s design for a spring watch—five years before Huygens’s announcement—vindicating the homely scientist.
Interest in Hooke’s contributions to science has revived in recent years, and these include a footnote in the history of calculus—specifically, the catenary and its importance to architectural arches. Thanks to his early apprenticeship to an artist, Hooke was a gifted draftsman, and his architectural bent proved useful when the Great Fire of London destroy
ed much of the city. It was in the process of rebuilding St. Paul’s Cathedral in 1671 that Hooke “rediscovered” the secret of the catenary.
Alas, Hooke was a bit too clever for his own good. He announced his solution to the problem of the optimal shape of an arch to the Royal Society, but he never published it. Instead, four years later, he published an encrypted solution in the form of a Latin anagram in the appendix to his treatise, Description of Helioscopes. It did not attract much notice. Finally, in 1705, the executor of his estate published the anagram’s solution: “As hangs the flexible chain, so inverted stands the rigid arch”—or, if you want to be all Latinate about it: Ut pendet continuum flexile, sic stabit contiguum rigidum inversum.
Had Hooke been less secretive about his discovery, he might have received credit sooner for his solution to the problem of the catenary. Instead, a German mathematician named Johann Bernoulli found the solution independently and announced it in 1691. Johann was one of eight gifted mathematicians and physicists in the legendary Bernoulli family. They were a virtual dynasty during this period. The Calvinist family originally hailed from Belgium but fled to Switzerland to escape Catholic persecution. There the family patriarch, Nicolaus, made his fortune as a spice merchant.
Nicolaus had intended that his son Johann take over the family business. Alas, Johann failed miserably as an apprentice in training and opted to study medicine at Basel University instead. In between his studies, he and his older brother, Jakob, began collaborating on the study of this shiny new mathematical tool called calculus and were among the first to apply it to various problems. Eventually, Johann switched from medicine to math, and thus began a series of nasty sibling rivalries that rippled through the Bernoulli family tree for decades.
The brothers Bernoulli were highly competitive, fought constantly—their letters to each other are filled with heated insults and strong language—and always sought to outdo each other when it came to posing mathematical challenges. (This practice of issuing challenges was all the rage back then among mathematically minded sorts.) The fact that Jakob had trained his younger brother made it difficult for him to accept Johann as an equal. Johann, in turn, hated to be outdone; he was even jealous of his own son Daniel, banning his offspring from the house when Daniel won a math contest at the University of Paris that Johann had also entered. Nor was he averse to a spot of plagiarism: He once stole one of Daniel’s papers, changed the name and date, and claimed it was his own work.
That constant bickering might have been ruinous to harmonious familial relations, but it seemed to fuel the Bernoulli brothers’ mathematical creativity. It was Jakob who set forth the problem of the catenary: determining the precise mathematical shape formed by a hanging chain. Nearly fifty years before, Galileo theorized that it formed a parabola, but this was disproved in 1669, leaving the matter open to debate.49 Johann Bernoulli, Leibniz, and Huygens all responded with their solutions within months, beating poor Hooke to publication. In modern calculus, it is possible to find the solution of the optimal shape for a hanging chain via a minimization problem, because the goal is to minimize tension. In contrast, finding the strongest shape for an arch is a maximization problem, because we wish to find the shape with the most compression forces.
There is yet another quirky feature of the catenary: It is related both to exponential growth curves and to exponential decay curves, according to Paul Calter, a retired math professor from Vermont and author of Squaring the Circle: Geometry in Art and Architecture. We’ve already seen how both curves apply to computing compound interest, for example, or to population dynamics; the only difference is a minus sign in the relevant equation for exponential decay. Calter points out that if you fit the two curves together, you get a catenary. So in the classic catenary shape, the descending portion of the curve behaves like exponential decay, while the rising portion of the curve exhibits the characteristics of exponential growth. And this shape, when inverted, forms a stable arch.
In 1696, Johann countered with a challenge to solve a particularly knotty conundrum: the problem of the brachistochrone. The word derives from the Greek words brachistos (“shortest”) and chronos (“time”). Johann cheated a little, having already solved the problem himself, but the challenge was deceptively simple on the surface. Assuming two fixed points, one higher than the other, what shape would a curved path between those points have to be for a rolling ball to reach the lower point the fastest? (In fine physics tradition, this problem assumes constant gravity and ignores friction.)
You might be tempted to dredge up a bit of long-forgotten knowledge from your high school geometry class and suggest that the shortest distance between two points is a straight line—ergo, a straight line is the fastest possible path. Resist that temptation. We are dealing with a curve in this instance. Galileo proposed in 1638 that the curve would be the arc of a circle; he, too, was mistaken. If we actually performed the experiment, it would quickly become clear that the steeper the curve between the two points, the faster the ball will gain speed.
Technically, this is a minimization problem: We are attempting to find the least possible amount of time it takes for the ball to descend. Yet because there is more than one quantity that is varying, the solution involves considering each and every possible path between the two points—truly a job for calculus. The solution is a cycloid, which is the curve created by a point on the rim of a wheel along a straight line.
Turn that path upside down—as with the inverted catenary curve that gives one the optimal shape for a stable arch—and you will get the path of fastest descent. You can test this result by building two tracks: one shaped like a cycloid, the other shaped like the arc of a circle, for comparison. Now roll two balls down each track simultaneously. The one on the cycloid path will reach the bottom first. Nor does it matter where one starts the ball along this curved path; it will still arrive at the bottom in precisely the same amount of time.
Five individuals solved the brachistochrone problem posed by Johann Bernoulli correctly: Johann himself; his brother Jakob; Guillaume François Antoine, Marquis de l’Hôpital; and the two founders of calculus, Leibniz and Newton. Newton was working as Master of the Mint at the time and received the challenge after a long day’s work. In general, Newton was loath “to be dunned [pestered] and teased by foreigners about mathematical things.” But the story goes that he was sufficiently intrigued by the problem that he stayed up until four A.M. until he solved it. All in all, it took him twelve hours. He submitted his solution anonymously to the Royal Society, but Johann was not fooled, claiming, “I recognize the lion by his print.”
In the process of uncovering the solution to this puzzle, the calculus of variations was born; Johann’s student, Leonhard Euler, refined his mentor’s techniques in 1766 and coined the term. This is calculus with an infinite number of variables. One would normally try to find the optimum value for a single variable (x), but in the case of the brachistochrone problem, it is necessary to integrate over all possible curves to find the optimal solution—selecting one curve from an infinite number of possibilities.
BARCELONA’S BIZARCHITECT
Visitors to Barcelona invariably become enchanted with the city’s unique architecture. In particular, one can see myriad catenary shapes in buildings designed by the great Catalan architect Antoni Gaudi y Cornet. There has never been an architect quite like Gaudi, who relied less on traditional geometric shapes and more on complicated hyperboloids and paraboloids—and of course, on the catenary. His designs also incorporate brightly colored mosaic tiles and whimsical ornamental touches like the multicolored mosaic dragon fountain at the main entrance of Parque Güell.50 At least one writer has described the flamboyant Gaudi style as Gothic Psychedelia, or bizarchitecture.
Descended from a family of coppersmiths, Gaudi enrolled at the Escola Tecnica Superior d’Arquitectura in Barcelona after a two-year stint in the military. His father sold the family property to pay for his son’s education, and Gaudi further earned his keep b
y working for Barcelona builders. Gaudi was not the most stellar student; he was too quirkily eccentric for that. One project involved the design of an entry gate to a cemetery. Gaudi embellished the basic blueprint by drawing a hearse and a smattering of mourners to set the mood, but forgot to draw the actual gate he’d been assigned to design. He received a failing grade. But two of his subsequent drawings received the highest marks, and eventually he earned the official title of architect. “Who knows if we have given this diploma to a nut or a genius? Time will tell,” sighed Elies Rogent when he signed Gaudi’s diploma in 1878.
Even today, Gaudi’s work is not universally admired, and in his early career, his designs were so bizarrely original that more often than not, he was ridiculed rather than praised. (George Orwell purportedly loathed Gaudi’s style when he lived in Barcelona during the Spanish Civil War.) A select few recognized the signs of genius. He soon found a patron in wealthy industrialist Eusebi Güell and began building his reputation as a rising young architect. The Gaudi of this period cut a striking figure, with his blond hair, blue eyes, and ruddy complexion—unusual for someone of Mediterranean descent. He augmented this with the most fashionable of clothes and a carefully groomed beard. In short, he was a bit of a dandy, although later in life he renounced such frivolities.
Gaudi also had a nasty temper and could be incredibly stubborn when it came to his craft. Take his design for Casa Batlló, which included every last detail, right down to the furniture. It was a truly innovative renovation, showcasing the architect’s signature style, with balconies that appear to move and a large cross crowning an “undulating roof.” Unfortunately, the owner of the house (Josep Batlló) had this silly notion that his furniture and aesthetic tastes should be taken into consideration, as he would be the actual occupant.
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 18