Shape of a balloon I have a balloon consisting of thin rubber and willed with helium or air under some pressure greater than 1 atmosphere.  The balloon is tied off at the bottom.
Without the complication of the tie at the bottom, and neglecting minutia such as air pressure gradients, the balloon would be a sphere. But we always picture party balloons as rounder at the top and a bit more sharply curved at the bottom.
What dictates the shape (is it the additional weight of the bottom knot, or a string attached that keeps the balloon static in position?) and what is the equation of the shape.  
 A: My guess is that, leaving aside the tying off knot, the latex composing  the ballon is not spread evenly. Balloons are made by dipping a spherical shape/mold into a vat of liquid latex and then lifting it out and allowing it to dry, so latex will flow downwards.
Also, they might increase the thickness of the balloon at the neck, to take the strain of tying the knot.

Latex balloons being manufactured.
As for equations of shape, I honestly have no idea of how these  could be derived from first principles, sorry.
A: For obtaining the shape, I would write the equilibrium of all forces: Archimedes' force and the reaction forces which are the string pulling down the balloon and the elasticity forces issued from the balloon membrane. In any case, the shape will not be a sphere, because the forces that are acting have no spherical symmetry.
A: The shape of the balloon is not technically a sphere. However, it is affected by the amount of air put into it. If you put less it might result in ball shape with something point upwards. Vice versa, more will affect in a sphere with a longer tip.
