Harmonograms, Spirographs, and Lissajous Figures

Encyclogram draws harmonograms, spirographs, and Lissajous figures. The decaying motion of the plot fills in the shapes with their spiralling-in echo. Encyclogram can also draw the curves in varying colors against a black background, resulting in breath-taking works of art that can be as beautiful as fractals. See the gallery of examples.

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.

640x480, 800x600, 1024x768

Frequency generally has the greatest effect, controlling the number of lobes. Amplitude controls the lobe sizes. Phase controls lobe orientations. Spiro gangs the x and y amplitude and frequency controls, and sets the phases for symmetric spirographs. Decay makes the curve spiral inwards. Color sets a black background, and draws the picture in color. WARNING: selecting the color option slows down the picture drawing. Thick draws a thick line. This also slows the drawing speed.

Harmonograms: select the decay option.

Spirographs: select the spiro option. Leave the phases alone for true spirographs. You can change them for interesting effects, but the pictures won't then generally be spirographs.

Lissajous: unselect spiro and decay. Set one amplitude on each side to zero; adjust the sliders in the non-zero groups (e.g. if you set the top sliders each side to zero, adjust the bottom three sliders).

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Click on the above link to download a zip file of the classes and HTML.

After conducting several searches, I believe this applet is unique in being the only one that makes it easy to draw spirographs and Lissajous figures, as special cases of harmonograms. The most popular applet for drawing so-called spirographs actually approximates them with overly long straight line segments, and has a few other weaknesses which E/g remedies. There are very few applets drawing these kinds of curves.

E/g will appeal to mathematicians and physicists, as an example of the resultant trace of perpendicular damped harmonic motions; to artists and graphic designers as a tool for exploring patterns in curves; and to almost everyone as a fun visual toy. Web designers may add it to their sites for visitors' use and entertainment. Teachers especially could use it to illustrate aspects of physics, maths, and art.

Future Plans:

  • Add an animation mode.
  • Display the curve's equation?
  • Allow to change the rate of decay?
  • Allow to select some preset curves?
  • Allow variation of the drawing increment?
  • Document the mathematics of harmonograms etc.
  • Allow initial display to be set by external parameters.
  • Create a version w/o explicit controls, e.g. for page decor.