Mylar Balloon and Associated Geometro-Mechanical Moments

Starting with identifications of the very fundamental geometric characteristics of a Mylar balloon such as the profile curve, height, volume, arclength, surface area, crimping factor, etc., using the geometrical moments In(x) and In, we present explicit formulas for them and those of the mechanical moments of both solid and hollow balloons of arbitrary order. This is achieved by relying on the recursive relationships among elliptic integrals and the final results are expressed via the fundamental mathematical constants such as p, lemniscate constant (sic), and Gauss's constant G. An interesting periodicity modulo 4 was detected and accounted for in the final formulas for the moments. The principal results are illustrated by two tables, a few graphics, and some direct relationships with other fundamental areas in mathematics, physics and geometry are pointed out.

Automated summary

In this paper, explicit formulas for the geometric characteristics of a Mylar balloon and the mechanical moments of both solid and hollow balloons are presented. The formulas are derived using recursive relationships among elliptic integrals and expressed in terms of fundamental mathematical constants such as pi, lemniscate constant, and Gauss's constant G. The periodicity modulo 4 in the final formulas for the moments is investigated and illustrated using tables, graphics, and connections to other fundamental areas.