Eiffel Equation

When Alexandre-Gustave Eiffel designed his famous freestanding 986-foot iron tower in Paris in the late 19th century, he did so without modern science or engineering. But mathematicians have long suspected that an elegant logic lies behind the monument’s graceful shape. That conviction led engineer Patrick Weidman of the University of Colorado on a quest to derive the equation behind the tower’s curvature. Initially frustrated, Weidman’s eureka moment came when he found a long-overlooked memo written by Eiffel in 1885. The document gave Weidman the insights he needed to work out the mathematical equation that describes the tower. Its shape is an exponential function of the natural logarithm—a mathematical curve frequently found in nature. “Eiffel figured out that with a special shape you can have zero wind load on the diagonal elements,” says Weidman. “That was a fantastic new discovery in construction.” His innovation not only reduces weight but also has a distinct aesthetic advantage by cutting out a heavy network of diagonal trellises, Weidman says. “It would look ugly if it had all these things going across the tower.”