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# Calculus and the Eiffel Tower: By Zara Hood

Describing the Eiffel Property
by

## Zara Hood

on 22 April 2015

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#### Transcript of Calculus and the Eiffel Tower: By Zara Hood

Calculus and the Eiffel Tower: By Zara Hood
Building the Eiffel Tower
Gustave Eiffel, a French civil engineer and architect, erected the Eiffel Tower in 1889. The tower is 300 meters (986 feet) tall; the highest accesible level is the observatory at 915.7 feet. The Eiffel Tower is considered one of the most mathematically perfect structures.
Graph of the Eiffel Tower
The Eiffel tower can be graphed using the functions e^(-x) and
-e^(-x).
Eiffel Property
At any height on the tower (AA in Figure 1), the tangent lines to the tower (correspond to the supporting forces of the tower) must pass through the center of mass, x, of that part of the tower above AA.
Bibliography
Sources
Sources Cont.
•Sundaram, M.M. and GK Ananthasuresh. "Gustave Eiffel and His Optimal Structures." IISC. 1 Sept. 2009. Web.
•Weidman, Patrick and Pinellis, Iosif. "Model Equations for the Eiffel Tower Profile: Historical Perspective and New Equation." UC Boulder. 13 Feb. 2004. Web.
•"The Exponential Eiffel Tower." Ucdenver.edu. Web. 17 Apr. 2015.
•"Elegant Shape Of Eiffel Tower Solved Mathematically By CU-Boulder Prof." University of Colorado Boulder. 5 Jan. 2005. Web. 17 Apr. 2015.
Wind Resistance
Eiffel's Hypotheses:
•Equilibrium of Moments
•Constant Axial Stress
If a strong wind hits the tower from one side, the force must be evenly distributed among the beams to counterbalance the wind load.
Finding the Center of Mass
To do this, you must find the lines tangent to the curves.
Find the derivative of the equations and then plug in point.
e^(-x): y= —e^(-1)x + e^(-1)(2)
—e^(-x): y= e^(-1)x — e^(-1)(2)
So how is Calculus involved?
Eiffel wanted the tower's design to wiithstand all weather conditions, taking into most account wind resistance. The triangular shape of the tower provides stability, but is not a perfect geometric shape. Rather, the tower is made with curves, incorportaing calculus.
Optimization
The Eiffel tower is considered an optimal structure.
•eliminate trellis element
•more interior space
•less surface area
•wind passes through
Continuous Model of the Eiffel Tower
Weidman and Pinellis found a nonlinear integro-differential equation to represent the profile of the Eiffel Tower and illustrate Eiffel's concern for wind load.
Images
•Eiffel Tower Paris Night Wallpaper. Digital image. Architectboy.com. N.p., n.d. Web. 17 Apr. 2015.
•Gustave Eiffel Building the Eiffel Tower. Digital image. Bibliography.com. N.p., n.d. Web. 17 Apr. 2015.
•All diagrams are from sources already cited.
Full transcript