Why does the Fizeau measurement of the speed of light use a sawtoothed gear instead of a single gap? In his measurement of the speed of light, why did Fizeau use a sawtooth gear with $n$ teeth and $n$ gaps? Why not use only one gap, so when light passes through the gap we will know that it is that unique gap and we wouldn't have to calculate $\theta=2n/2\pi$?
The experiment looks, schematically, as follows:

Image source
 A: The more teeth you have the slower the rotation speed of the wheel can be.  He had 720, with a single tooth you would need to rotate the thing 720 times faster.  The description of the experiment says that the wheel could rotate at hundreds of Hz, so a single-tooth wheel might need to spin at, perhaps, 100,000Hz or more.  This would be challenging to put it extremely mildly.
In fact, based on a 16km path, the lowest rotational speed is $3\times 10^8/(16\times 10^3\times 720) \approx 26\:\mathrm{rps}$ for a 720-tooth wheel.  This is perfectly mechanically viable, although it would have been fairly demanding in 1851.  720 (or in fact 360, see below) times this would be extremely challenging now.  Thanks to Emilio Pisantry for the numbers.
(A single-tooth wheel is also asymmetric -- it's a cam, really -- which would make things catastrophically worse: a two-toothed wheel is less mechanically insane.)
A: Basically it is because the speed of light is so very very high which requires the wheel to spin very fast and/or have many narrow notches.
Fizeau used the wheel to "time" light as it travelled a distance of 16km. Light passing through one notch on its outward journey was blocked by the adjacent tooth when it returned, after the tooth had moved a very small distance. Even though the wheel rotated a few hundred times per second, it was still necessary to have 720 teeth in order to get the experiment to work. 
If the wheel had only one narrow notch, it would still have to rotate at the same speed as if it had 720 teeth, but only $\frac{1}{720}$ of the same amount of light would get through when the notch is aligned for the outward and returning light. This makes it much more difficult to distinguish at what point this light is completely blocked out, because there is so little contrast between "light" and "dark". 
Fizeau was able to know that the light was blocked by the adjacent tooth because he increased the speed gradually, observing the light get dimmer. When the transmitted light reached a minimum the first time he could be sure it was blocked by the adjacent tooth. When the 2nd minimum came because of the next tooth, this was when the wheel spun 3x as fast. The big difference in speed made it possible to be sure which tooth was blocking the light. 
Reference : wikipedia article
see also How did Fizeau make his famous speed-of-light experiment? and How did Fizeau control the cog's rotations per second? 
A: It is necessary for the sawtooth pattern to be regular and having some amount of rotational symmetry because the wheel needs to be spinning with uniform angular velocity.
If you have a saw-wheel with only one tooth, then it's very difficult to get the wheel to spin uniformly! It will require a variable input to counter-act the rotational torque which arises from spinning the irregular wheel.
For example, having worked with saw mills and seeing spinning saw blades in action, the blades need to be discarded after they hit a nail. Why? Because if a single saw-tooth is damaged, then the saw-blade begins to oscillate and vibrate, and the saw no longer cuts straight.
