February 1, 2005
Iosif Pinelis, a professor of mathematical sciences, first became intrigued by the problem in 2002, when Patrick Weidman, an associate professor of mechanical engineering at the University of Colorado at Boulder, visited Michigan Tech (MTU). Prof. Weidman presented two competing mathematical theories, each purporting to explain the Eiffel Tower’s elegant design.
One, by Christophe Chouard, argued that Eiffel engineered his tower so that its weight would counterbalance the force of the wind. According to the other theory, the wind pressure is counterbalanced by tension between the elements of the tower itself, Prof. Pinelis said.
Chouard had developed a nonlinear integral equation to support his theory, but finding its solutions was proving difficult. “Weidman and the mathematicians whom he had consulted could only find one solution, a parabola, of the infinitely many solutions that Chouard’s equation must have,” Prof. Pinelis said. As anyone who has survived high-school geometry can quickly testify, the Eiffel Tower’s profile doesn’t look anything like a parabola. Prof. Weidman asked MTU mathematicians if they could come up with any other solutions.
Prof. Pinelis went back to his office and soon found an answer confirming Prof. Weidman’s conjecture that Chouard’s theory was wrong. It turns out that all existing solutions to Chouard’s equation must either be parabola-like or explode to infinity at the top of tower.
“The Eiffel Tower does not explode to infinity at the top, and its profile curves inward rather than outward,” Prof. Pinelis notes. “That pretty much rules out Chouard’s equation.”
Prof. Weidman then went to the historical record, and found an 1885 memoire delivered by Eiffel to the French Civil Engineering Society affirming that Eiffel had indeed planned to counterbalance wind pressure with tension between the construction elements.
Using that information, Prof. Weidman and developed an nonlinear integral-differential equation whose solutions yielded the true shape of the Eiffel Tower. That shape is exponential.
The work by Prof. Weidman and Prof. Pinelis, “Model Equations for the Eiffel Tower Profile: Historical Perspective and New Results,” has appeared in the French journal Comptes Rendus Mecanique, published by Elsevier and the French Academy of Sciences.
“The funny thing for me was that you didn’t have to go into the historical investigation to disprove a wrong theory,” Prof. Pinelis says. “The math confirms the logic behind the design. For me, it was more fun to go to the math.”