With aircraft, there are 6 degrees of freedom (DoF) we want to control (roll-pitch-yaw, and x-y-z), but for hovering vehicles (i.e. rotorcraft) we can get away with controlling two degrees of freedom (x-y) by combining the 4 other degrees (roll-pitch-yaw + z).
There are many ways of controlling those four DoF, typically:
- helicopters, which use variable pitch blades and some combinations of main and tail rotors, to result in a combination which results in control over roll, pitch, yaw, and thrust
- tricopters, quadcopters, hexacopters, octocopters, etc..., all of which use their blades to control the roll, pitch, yaw, and thrust.
Somewhere it is inescapable that for stabilized flight on all four DoF you must have the ability to control each DoF. Mathematically, the quadcopter provides the lowest number of simple actuators that accomplish the job. Each actuator can work by simply speeding up or slowing down, whereas helicopters and tricopters require some other kind of actuation, usually a complex one. It turns out from a pragmatic viewpoint speeding up and slowing down motors is much more robust than changing angles of spinning blades.
Formally, it looks like this:
So quadcopters reign over helicopters and tripcoters because they're simple. Each axis is controlled by a certain combination of motor speeds and all axes are independent, i.e. when you increase thrust you don't have to also worry about changing the pitch.
So why not hexas, octos, etc...?
Hexas can lift more, but provide no additional safety margin because there is no way to stabilize all four axes when an arbitrary motor goes out (there are certain configurations which can continue to fly if one motor out of a set of four fails, but cannot fly if either of the other two fail).
Hexas also are less efficient, since they use smaller propellers. (Without diving into blade theory, and simplifying somewhat, the most efficient blade is a single blade which is infinitely large and moving infinitely slowly.)
Hexas also have 6 motors for four degrees of freedom, so when they're all flat you get what is called an "overconstrained" system. You can actually angle them to interesting effect, e.g. CyPhy Work's LVL 1 drone.
Octos theoretically can continue to fly if any arbitrary motor fails, but in reality, you'll likely encounter blade stalling effects or other pathologies that will not allow the octo to sustain flight. This happens because the typical octo is loaded past the point where 4 motors can carry it safely. However, certain high-value vehicles, such as those carrying expensive cameras, are octos because it's cheaper to over-spec the octo's batteries and motors than it is to replace a $50k camera.
Octos are also less efficient, for the same reason as hexas.