If you search the internet for the most aerodynamic shape, you will find a couple pages saying it's a teardrop. This seems pretty reasonable. It's about the shape of a wing, although wings aren't designed only for minimal drag.
But nothing really authoritative. No indication that any college kid ever spent thousands of hours with a wind tunnel trying all possible permutations. Why not?
So then you scour the web for some kind of mathematical definition of a teardrop, and you eventually come across someone suggesting a sphere and a sin wave. I was familiar enough to know that the math for the two objects are very closely related, and quickly created a model that satisfied me.
While I understand that the tail end is more aerodynamically significant, I wondered why more of a point on the nose wouldn't help. I wondered what a full sin wave would do. That looked pretty good. I thought maybe if you took a sphere or cylinder and used the flow paths around that, you would get the perfect shape.
Then I watched a couple excellent videos of flow paths around a cylinder and a plank, and realized that if the plank's flow paths weren't similar to those of the sphere, I would need to build this shape in an iterative process, using the new shape's flow paths to calculate the next shape. Looks like that would turn out like a sin wave.
This thought process was all based on the air particles very close to the object. Surely the air particles farther away would also appreciate a sin wave? I was disappointed that a shape more complex than a sin wave would probably be necessary. Then I contemplated the flow paths of those particles. They would get something nicely close to a sin wave.
I don't think a full sin wave would be practical often, because a small change in angle of attack could be bad. So perhaps an average between a sphere and a sin wave?
Also, the sin wave comes to an extremely thin and long end. I suspect it would be more practical to chop off the bottom 5% or so of the sin wave.
I hope to build a wind tunnel out of cardboard and box / window fans, and a drag balance and do some testing with clay.
The aerodynamic differences between a sphere and a cylinder would be interesting. Particles around a sphere get spread out as they're cramped over the object. Around a cylinder they all take the same parallel path.
There is an interesting page on boat hulls which mentions that no matter how lovely your shape, Osborne Reynolds says if it's long enough, your laminar flow will turn turbulent. I'm very curious how far that is in air. I'm hoping far.
Finally found an equation defining airfoils: http://en.wikipedia.org/wiki/NACA_airfoil#Equation_for_a_symmetrical_4-digit_NACA_airfoil
Yup, "Aerofoils with pointed noses have cost several lives because they stall totally and suddenly." So that appears to be the only reason wings are teardrop shaped and not pointy: They suddenly fail to cause lift. This page also implies that airfoils are based on parabolas.