Closed Geodesics on the Mylar Balloon Shape

In 2006, Alexander proved a result that implied for the mylar balloon shape, if n is the number of times a closed geodesic winds around the axis of rotation and m is the number of times the geodesic oscillates about the equator, then n/m ∈ (1/ √2, 1] . In this paper, we will provide a simpler more direct proof of Alexander's result for the mylar balloon by using sharp estimates of certain improper integrals. The mylar balloon geodesics suggest a network of reinforcing fibers creating an isotensoid system that is lightweight, compactly foldable, and easy to deploy, making it an ideal candidate for an inflatable terrestrial or space habitat. Other applications of geodesics, including geodesic domes, aerodynamic decelerators, and the production of salami are also discussed.