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# Calculus in Architecture

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by

## Mackenna Johnson

on 21 November 2013

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#### Transcript of Calculus in Architecture

BY: Kelly Wizer and
Mackenna Johnson

For Example,
The dome of the Taj Mahal in India, built between 1632-1653 by Shah Jahan, involves complex integral calculus to determine the exact shape of the dome.
Differentials and Integrals
Differential Calculus: An important part of architecture is figuring how to minimize or maximize an aspect of the building. For example, to maximize window size in a building, an architect is finding a maximum by using a derivative (in other words, where the slope is zero) in order to create the most window space possible based on the proportions and aesthetics of the building.
Why is the center important?
Well, friends, this is a valid question. The ability to find the center ties back into what we said earlier about the way the Eiffel tower is molded from wind. If one knows where the mass is centered, he/she can more effectively balance the tall structure using the wind as an ally instead of an obstacle.
La Tour Eiffel!!!1!
When Gustave Eiffel built the tower, he did not actually use calculus, but instead based the dimensions of the tour on practical experience. Despite this fact, though, the Eiffel Tower is one of the, if not
the
, most mathematically perfect structures on the planet. Eiffel took force of wind into account most often; he built the tower to be highly durable in even the most extreme weather. Part of this comes from its almost triangular structure, with its wide base that narrows into a point on the top. The tower is also constructed in a very unusual manner. Eiffel utilized many different kinds of beams (ie circular, square, etc), while most architects use the same sort of beam.
The Graph
Application: The Eiffel Tower
Yes, ladies and gents, there most certainly is. Architects could use simple geometry if buildings were made of ideal squares and triangles and such. However, the vast majority of buildings involve curves and strange angles. That, then, is where calculus comes in.
Calculus in Architecture
There's Calculus in Architecture?!
Integral Calculus: If an architect wanted to find how much heat a building loses based on temperature variations throughout a day, he/she would have to graph heat loss vs. time and find the area under the curve that creates.
The Eiffel Tower was erected in 1889 by Gustave Eiffel. The iconic and romantic structure stands 1,063 feet tall in the heart of Paris. The highest accessible level of the tower is the observatory, 915.7 feet high. It opened to the public just a few short years before the World's Fair in Chicago.
The Eiffel Tower, surprisingly enough, can be graphed by two simple equations:
e^-x
-e^-x
Here's what it looks like:
More Math Yay
Where f(x) is the half-width of the Tower at height x, f0 is the half-width of the Tower at the base of the structure, and w(x) is the maximum wind pressure the Tower can withstand at height x.
How to Find the Center of the Eiffel Tower
It is important to be able to find the center of the Eiffel tower so that one can find where the mass of the tower is focused. To do that, we must find the tangent lines of the curves we previously graphed.
Graph of the Tangent Lines
Works Cited
http://www-math.ucdenver.edu/~bennethm/courses/calc3/EiffelTowerProject.pdf
http://www.mecheng.iisc.ernet.in/~suresh/journal/J56EiffelOptimalStructuresCompressed.pdf
http://www.sciencedaily.com/releases/2005/01/050106111209.htm
http://plus.maths.org/content/shaped-wind
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