The Calculus Diaries - Jennifer Ouellette [79]
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