There are several factors to consider when talking about the flight of a body moving dynamically though air.
An arrow achives the majority of its stability in two major ways. The first is through forcing its center of pressure to be behind its center of gravity. The center of pressure on a body is the point where all of the aerodynamic pressure fields may be represented by a single force vector with no moment. This condition yields stable flight, meaning the arrow will not overturn during flight due to aerodynamic forces. In this regard, there is little gained using 4 vanes instead of three, since the center of pressure for either scenario will be adequately aft of the center of gravity to be stable.
The second is through gyroscopic spin. In this case, spin is induced by the offset of the vane relative to the axis of the shaft. This rotation aids in stability by conservation of angular momentum. In this regard, 4 vanes offer superior stability over three since the additional vane add an additional force to induce rotation faster.
Consider the major aerodynamic forces acting on such a body throughout the duration of the flight. Lift and Drag. To stay relevant to this discussion, consider lift only to be a function of velocity, angle of attack and surface area. And drag to be a function of velocity and surface area, and surface roughness. That is, as velocity, angle of attack, surface roughness, and surface area change, there is a corresponding change in lift and drag.
For an arrows flight, the lift (in the technical sense of the term) provided by a set of vanes is negligible since the angle of attack is 0* (that is the velocity vector is essentially congruent with the axial body vector). However, vanes are imperative to an arrows flight being dynamically stable. Vanes provide a small restoring lift force even at small angles of attack. This is important to overcome turning momentum. In this regard, 4 vanes offer superior stabilization by providing a larger total surface area.
Drag is a non-negligible force acting on the arrow and has to be accounted for. Drag increases with surface area, velocity and surface roughness. There are multiple types of drag, but here I will do some hand waving and throw them all in one bucket. If someone is interested in a more in depth explanation I can expand. With 4 vanes, you have a ~33% increase in surface area over 3 vanes. This will result in an increase in surface drag. There is also an increase in interference drag (drag cause by interference of the airflow over the body and vanes at the junction) with 4 vanes.
Someone touched on surface roughness above, I will just mention that it does affect drag. Sometimes it is even desirable to increase surface roughness or introduce something to obscure the airflow over the vane to reduce surface drag. This is called tripping the flow. It is however not of great value to this discussion considering the velocity at which the arrow is traveling.
With all of this and more in consideration, I shoot 3 vanes. 3 quality vanes (99% of the time) are more than adequate to stabilize the arrow and the ~25% reduction in drag gained by only having 3 vanes is of much more value than the small amount of stability gained by 4 vanes.
Just my $0.02
-Nathan
LOL for me there performance has tightend my groups by an inch. Worth the extra two dollars for me. Plus fletching arrows is fun!!!
