How many ways can one possibly spin an object at the end of a string? I had a job that required me to carry a set of keys attached to a chain for most of the summer. I found that I could spin my keys at the end of their chain in two ways: in a circle, and in a figure-8. My hypothesis was that I could add another loop to the figure-8, but I found that i was unable to spin my keys in this fashion. 
How many ways could I possibly spin my keys, such that the keychain remains taut at all times? Have I discovered the only two ways of doing it, or are there more ways of spinning them? If there are more ways of spinning my keys, then what are they?
 A: Very interesting question!
If we think of the keys hanging like a pendulum and your hand not moving then since the lenght of the pendulum is fixed you can only have the same periodicity in the oscillations on the x-axis and the y-axis, so you only can perform segments, circles and in general ellipses in that configuration. So the ratio between the periods of oscillation in both axis can be expressed as $1:1$

Your figure 8 is the result of your hand moving and giving some extra energy in some direction just to give the "pendulum" a different periodicity in the x-axis than the y-axis. In fact the figure 8 is a product of a ratio $1:2$ between the periods.

In general you should be able to do any Lissajous curve with your keys. For example, here you have one made using a $4:5$ period relation between the oscillations in both axis:

So, why you can't do any Luissajous curve you want with your keys and only the ones corresponding to $1:1$ and $1:2$? Well, I don't know the true answer and I also would like to know if someone here can give a rigorous calculation but my guess is that there is nothing that forbids this to happen. The problem is that the hand movements you need to perform to create those curves are either too fast, too unnatural, too specific (with low tolerance in terms of doing it a bit differently) or a combination of those just to make it look almost imposible to do in practice.
