# The Curious Case of the Eiffel Tower

Debate has simmered among engineers over just why Gustave Eiffel designed his famous tower the way he did. Now it appears that the matter has been put to rest, thanks in part to the equations of Michigan Tech mathematician Iosif Pinelis.

Eiffel Tower

Pinelis, a professor of mathematical sciences, first became intrigued by the problem in 2002, when mechanical engineering professor Patrick Weidman of the University of Colorado at Boulder visited Michigan Tech. Weidman presented two competing mathematical theories, each purporting to explain the Eiffel Tower's elegant design.

The first, by Christophe Chouard, argued that Eiffel engineered his tower so that its weight would counterbalance the force of the wind. Chouard had developed a complicated 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," Pinelis said. As anyone who has survived high-school geometry can testify, the Eiffel Tower's profile doesn't look anything like a parabola. Weidman asked Michigan Tech mathematicians if they could come up with any other solutions.

Pinelis went back to his office and soon found an answer confirming 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 the tower.

"The Eiffel Tower does not explode to infinity at the top, and its profile curves inward rather than outward," Pinelis notes. "That pretty much rules out Chouard's equation."

According to the second, longstanding theory of Eiffel Tower design, the wind pressure is counterbalanced by tension between the elements of the tower itself.

After examining Pinelis's equations, Weidman went to the historical record and found an 1885 letter from 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, Weidman and his colleagues developed an equation whose solutions yielded the true shape of the Eiffel Tower.

"The funny thing for me was that you didn't have to go into the historical investigation to disprove a wrong theory," Pinelis says. "The math confirms the logic behind the design. For me, it was more fun to do the math."