(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
sidetoside?
(iii) What happens if $a$ is equal to an irrational number? Will
the curve ever repeat?

