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Abstract:
In this paper, the very fundamental geometrical characteristics of the Mylar balloon like the profile curve, height, volume, arclength, surface area, crimping factor, etc. are recognized as geometrical moments ℐ

Keywords:
Mylar balloons, geometrical moments, elliptic integrals, beta and gamma functions, recursive relations, crimping factor, lemniscate constant

Abstract:
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 π, lemniscate constant ˜ω, 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.

Keywords:
solid and hollow Mylar balloons, crimping factor, geometro-mechanical moments, recursive relations, elliptic integrals and functions, gamma functions, Gauss’s and lemniscate constants