# The Gateway Arch - A Trigonometric delight

## Introduction

The Gateway Arch in St Louis was built as a monument to commemorate the pioneering spirit of the explorers who forged the westward expansion of the United States. As we stand under the great arch, we too will embark on a journey, an exploration into the realm of hyperbolic trigonometry as we discover the majesty of the catenary curve.

## Description

"Educate and inform the whole mass of the people... They are the only sure reliance for the preservation of our liberty"

Thomas Jefferson

St Louis, Missouri is home to the Jefferson National Expansion Memorial, a 91 acre park along the Mississippi River in the heartland of America. The park was established to commemorate among other things the Louisiana Purchase and is located near the starting point of the Lewis and Clark expedition. Instead of exploring the Pacifc Northwest of America, this location will be our starting point for a journey of a different kind, an exploration into the previously undiscovered world of the catenary curve. We will not follow the raging Mississippi, but rather the steady stream of hyperbolic trigonometry will be our guide to uncovering the riddle of the mysterious catenary curve as we go forth to explore the wonders of the magnificent Gateway Arch of St Louis. In the words of the great Thomas Jefferson, I hope this is a journey which will "educate and inform" as we enjoy together a stunning example of the use of mathematics in engineering and architecture.

Before we embark on our journey to discover the great arch, we must first explore the enigmatic catenary curve. In the May 1690 issue of Acta eruditorum, the Swiss Mathematician Jakob Bernoulli proposed the following problem..."To find the curve assumed by a loose string (or chain) hung freely from two fixed points". Galileo had earlier attempted to solve the problem, but his solution of a simple parabolic curve was proved incorrect in 1646 by the Dutch Mathematician and Scientist Christian Huygens. Finally in June 1691 three correct solutions to the problem were submitted, by Huygens, Gottfried von Leibniz and Jakob's brother, Johann Bernoulli. The curve which solved the problem was determined to be a catenary, given by the equation below (where "a" is a constant whose value depends on the physical parameters of the chain)

$y = a \, \cosh \left ({x \over a} \right ) = {a \over 2} \, \left (e^{x/a} + e^{-x/a} \right )$

The hyperbolic cosine function (cosh) is an indication that we are entering the realm of hyperbolic trigonomerty.

The graph below displays the shape of the catenary curve for different values of a.

The catenary curve has been immortalized in the Gateway Arch constructed in the Jefferson National Expansion Memorial and this is where our journey begins.

The Gateway Arch was built as a monument to the exploration of the United States west of the Mississippi but through the eyes of the mathematician it is a symbol of the exploration and discoveries made by many great mathematicians in the world of differential calculus. The arch was designed by Eeero Saarinen and Hannskarl Bandel. Construction of the arch was finally completed in 1965 and standing at 192m, it is the tallest man-made monument in the United States
The Gateway Arch itself is not the standard commom catenary but rather a more general curve known as an inverted weighted catenary whose base is thicker than its vertex. The monument opened to the public on June 10, 1967 and it is currently one of the most visited tourist attractions in the world with over 4 million visitors annually. A visitor centre is located below the arch, containing a musuem and two cinemas but the true mathematical explorer will take this oppurtunity to travel to the top of the arch. A tram in each leg of the arch takes us on a hyperbolic cosine path to the vertex and the observation area. From this lofty vantage point, our explorers can see up to 50km on a clear day.

The experience of travelling along a catenary is exhilarating but I believe the best vantage point from which to enjoy the majesty of this mathematical construction is from the ground. Here we can marvel at the wonders of mathematics in the architecture of this amazing structure. The beauty of the arch is only enhanced by understanding the mathematical nature of its construction, a wonderful example of the catenary curve.

The Gateway Arch,
Memorial Drive, St. Louis, Missouri, United States.

## Access information

Wheechair/buggy accessible:
No
Visiting times:
09:00-18:00 (08:00-20:00 in Summer)
Cost:
Tram Tickets (to top of Gateway Arch) $10 Adult,$5 Child
Access notes:

Best viewed from the Jefferson National Expansion Memorial.

Image of Gateway Arch

http://en.wikipedia.org/wiki/File:St_Louis_night_expblend_cropped.jpg

Image of 3 Catenary Curve

http://en.wikipedia.org/wiki/File:Catenary-pm.svg