Harmonograms are mathematically the sums of several harmonic motions in the x and y directions, decayed over time. If the decay is removed, and there are only two harmonic motions (sinusoids), one in x and one in y, then the graphs are Lissajous figures. If another harmonic motion is added to each axis, and they are all in a specific phase relationship, then spirographs can be generated. These are better-known as the result of rolling a (toothed) wheel around inside another wheel, with a pencil point through a hole in the rolling wheel.

You don't have to know any mathematics to use Encyclogram (though if you're studying trigonometry you'll find this applet is an interesting example of what can be done with sine curves!). Simply move the sliders around, and try the check boxes. Here's how it works:

For a quick start, click the Random button a few times; then check the Spiro box and click on Random a few times more. Check the Color button (and wait - it'll be slower).

At the left of the applet are two sets of x-pendulum controls; at the right of the applet are two sets of y-pendulum controls. x is the horizontal direction in the drawing area, and y is the vertical direction. Each pendulum (2 per axis) has an amplitude, frequency, and phase.