How is the shape of the blades of an air fan determined? Trial and error, or is there a theory behind it? What are they trying to maximize, volume of air dislocated per rotation?
Mostly trial and error - but in a computer.
Both of these are complex areas - and when they come to together you need a lot of expensive computers and some even more expensive engineers.
I would guess the Lift/Drag ratio of the blades is important. Lift = wind thrust (and speed), and Drag = friction and power loss.
Since the part of the blade moves slower the closer to the center, the angle of attack needs to increase to provide the same wind speed. Keeping an even velocity profile is probably important in order to minimize losses due to turbulence.
There is probably a lot to be done with the wing-tips that is not done due to aesthetic reasons. pffft.