| Summary: | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">I</mi><mi>n</mi></msub></semantics></math></inline-formula> and this observation has been used to introduce an infinite family of surfaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> specified by the natural numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>. These surfaces are presented via explicit formulas (through the incomplete Euler’s beta function) and can be identified as an interesting family of balloons. Their parameterizations is achieved relying on the well-known relationships among elliptic integrals, beta and gamma functions. The final results are expressed via the fundamental mathematical constants, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> and the lemniscate constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϖ</mi></semantics></math></inline-formula>. Quite interesting formulas for recursive calculations of various quantities related to associated figures modulo four are derived. The most principal results are summarized in a table, illustrated via a few graphics, and some direct relationships with other fundamental areas in mathematics, physics, and geometry are pointed out.
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