rwg>The (mostly impressive) Mma 6.0 doc gives, under applications of EllipticF,
"Parametrization of a mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):", followed by three definitions and a call to ParametricPlot3D that makes an "M&M". Inelastic mylar would crinkle at the equator, so the true solution would be more like two shallow paper cupcake molds.
No, that can't be right either. More efficient would be for the inward creases to be cusps and the outward ones rounded. But then the distance from the equator to a pole along a groove would be less than along a ridge, and the only obvious sink for the extra length would be the grooves wastefully curving inward past vertical. Real mylar balloons solve this problem by having the grooves change to ridges on the way up, which I'm happy to have predicted before confirming at the supermarket. But on what scale are these grooves and ridges? Don't they depend on the thickness of the mylar? I.e., wouldn't thinner mylar permit a greater number of smaller puckers? Might it be that the grooves in a 200' diameter "satelloon" (hopefully bearing a giant Happy Face and the legend, "Eat This, PG&E") be no deeper than a one footer? (I wonder if the Echo designers considered this presumably cheaper alternative before somehow figuring out how to make that moby sphere.) If the wrinkles don't scale, then they're infinitesimal in mathematically ideal mylar, and the Mathematica plot is (probably) right! Ideal mylar would be this strange, infinitely shrinkable but finitely stretchable material, as if composed of a triangular webwork of microscopic keychains. --rwg PS: Researching an accessible shape for p{eta(q^a),eta(q^b),eta(q^23)}, I guessed that p{eta(q^3),eta(q^5),eta(q^11)} might be nice because 11-3 and 11-5 both divide 24. After a one day Resultant and a three day Factor, 1.03 MB, 1091 terms, degree 256, smallest coefficient = 3^81.