Lissajous Figures

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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?