Arch Mathematics
In the spring of 1999 and 2001 I incorporated a series of
Mathematica projects or ``labs'' into a vector analysis course. These labs used
the technology of a Computer Algebra System (CAS) to further develop a
particular concept, solve an application, or bring together several
disjoint ideas. By far the most popular lab explored the mathematics behind the
Jefferson National Expansion Memorial, commonly called the St. Louis
Arch, or just the Arch.
The Arch was constructed to celebrate the westward
growth of the United States brought on by the Louisiana Purchase under
President Thomas Jefferson. Construction of the Arch began February 12,
1963 and was completed on October 28, 1965. (For more historical and
statistical information on the Arch see the national parks website at http://www.nps.gov/jeff/). Besides being a must-see
for any tourist visiting Saint Louis, the Arch has some nice mathematical
properties. The Arch has height of 630 feet and is also 630 feet from
base to base. The center of the Arch follows the path of an upside-down
catenary curve: $A \cosh(t/A)$. Cross-sections of the Arch are equilateral triangles,
the largest of these at the base has a side length of 54.1 feet, and the
smallest triangle at the top has a side length of 17 feet. In the lab, students
found a parameterization of the Arch and used this parameterization to compute
its surface area.
In this
talk I will go over a little of the mathematics involved to solve the problem
as well as the impact the project had with the students.