Synchronous motors are 100% dependent on line frequency so their speed cannot be controlled by varying the ac voltage.
Regular fans bought from a local supermarket use induction motors. ...
When I was still employed, one of the PhD scientists would connect a table-top 10A Variac to a geared synchronous clock motor to rotate his coating targets inside a physical vapor deposition (PVD) vacuum chamber at a variable speed. The motor looked like this:
The image above was copied from this link
His gimcrack procedure actually worked! The rotating target offered virtually no resistance, but the gear train offered enough friction to cause "slippage" between the aluminum rotor and the magnetic field that induced eddy currents in the rotor to produce a miniscule amount of torque. As the voltage was reduced, the aluminum rotor did slow down. Torque at the geared output shaft suffered of course, but he didn't care since his application required no torque. And of course the motor was no longer operating in synchronous mode so timing suffered too, not that it mattered for his application.
No amount of explaining would convince him that his procedure was "improper" so eventually I quit trying to explain why it was improper. Since it was working for him, why beat on that horse? It didn't take much to convince him (some blown fuses in the Variac) that he needed an isolation transformer before trying to operate his rig in a well-grounded chamber filled with an electrically conducting plasma, but that's another story.
The same "slippage" effect occurs with induction motors too, and in fact is necessary to induce a magnetizing current and torque in the rotor windings, via transformer coupling between rotor and the rotating magnetic field of the stator. Torque is produced only when the rotor speed is less than the stator field rotation rate (called the synchronous rate) so the rotating stator field "cuts" the conductors of the rotor, a phenomenon called slippage.The induced poles on the rotor are always displaced with respect to the rotating magnetic field produced by the stator windings. The more torque that is applied to the output shaft as a load (at the rated operating voltage), the more slippage occurs until the motor stalls when the rotating magnetic field no longer induces sufficient current in the rotor to produce enough torque to overcome the shaft load.
In an induction motor, torque is reduced and slippage is increased (rotor rotates slower) if you reduce the operating voltage, for example with a Variac. Unfortunately a rather large change in voltage is necessary to produce a modest reduction in speed because torque is also proportional to the square of the applied voltage. Motor efficiency also decreases with reduced voltage and the motor may overheat if heavily loaded. Of course at a lower speed, the wind load on a fan is also reduced, so I suppose it all works out in the end by some fortunate happenstance.
If it works for you, go fer it. And by all means try using a light dimmer. Or buy a fan with multiple speeds and sit back and enjoy the breeze, knowing you are not only electrically correct but politically correct too.