In tracing the circle $x=\cos(t)$, $y=\sin(t)$, both the $x$ motion and the $y$ motion repeat every $2\pi$. This means that we trace the same circle over and over and over again. If we change the frequency of one of the two motions, then we get a more complicated curve called a Lissajous figure. In this demonstration we look at curves of the form $$x=\cos(t); \qquad y=\sin(at).$$

(i) First set $a$ equal to a small integer, like 1, 2 or 3. These are the simplest kinds of Lissajous figures. (ii) Now set $a$ equal to a simple fraction, like $3/2$ or $5/2$ or $2/3$ or $3/4$. What happens? When $a=p/q$, how many times does the curve go up and down before it repeats? How many times does it go side-to-side? (iii) What happens if $a$ is equal to an irrational number? Will the curve ever repeat?