To make colorful harmonographs, click the "color" box and make sure that "decay" is checked. Then click the "random" button (or you can drag any sliders and watch as the image changes).

To make colorful spirographs, click the "spiro" box and the "color" box. The "decay" box can be checked or unchecked. Then click the "random" button (or you can drag any sliders, but leave the phases alone for true spirographs).

To make colorful lissajous figures, click the "color" box, but leave the "spiro" box and the "decay" box unchecked. Then drag any sliders, but set one amplitude on each side to zero.

The kind of patterns shown here occur in nature. For example, the moon traces out a curve as it moves around the earth. As the earth orbits the sun, the pattern gets more complex. And as the sun moves around the center of our galaxy, the path the moon takes becomes remarkably similar to the patterns you see here. Mathematicians study patterns like these to better understand and describe the complicated motions of planets, satellites, machines and weather systems.

What is the math behind these drawings?
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.
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.
* Harmonographs are the sums of several harmonic motions in the x and y directions, decayed over time.
* Spirographs have two harmonic motions in each axis, all in a specific phase relationship. (Spirographs can also be created by rolling a toothed wheel around inside another wheel, with a pencil point through a hole in the rolling wheel.)
* Lissajous figures have two harmonic motions, one in x and one in y, with no decay.
You can enjoy the encyclogram without knowing any mathematics, but it is interesting to explore the effect of each slider on each type of figure... and the encyclogram also shows what can be done with sine curves, if you're studying trigonometry.

Harmonographs 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:

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.

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 pirographs. 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.

Harmonographs: 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).

Harmonograph
Mathematically, a typical 4-pendulum harmonograph may be modelled by

x = ( Ax1 * sin ( Fx1 * t + Px1 ) + Ax2 * sin ( Fx2 * t + Px2 ) ) * dk
y = ( Ay1 * sin ( Fy1 * t + Py1 ) + Ay2 * sin ( Fy2 * t + Py2 ) ) * dk

where the As are amplitudes, the Fs are frequencies, the Ps are phases, and dk = exp ( -k * t ), for some constant k.

Spirograph
The spirograph is

x = ( R + r ) * cos ( t ) - ( r + O ) * cos ( ( (R + r ) / r ) * t )
y = ( R + r ) * sin ( t ) - ( r + O ) * sin ( ( (R + r ) / r ) * t )

which is a harmonogram, by setting the decay constant to zero, Ax1 = R + r, etc, and adjusting the phase offsets to transform sine to cosine and to switch + to -.

Lissajous
Lissajous is simply

x = sin ( n * t + c )
y = cos ( t )

which again, can be derived from the harmonogram by suitable choice of parameters.

A Lissajous figure is a path traced out in the plane by a particle each of whose coordinates are under simple harmonic motion. Such trajectories are often encountered in physics.

Lissajous figures are sometimes called Bowditch curves after Nathaniel Bowditch who considered them in 1815. They were studied in more detail (independently) by Jules-Antoine Lissajous in 1857.

Harmonographs
Typically found in science museums, there are two basic forms: one comprises a large, heavy, rectangular platform suspended at all four corners by wires. The platform can be swung and twisted, and it can wobble forwards and backwards, side to side, and rotate a bit too. The other basic form comprises two or more pendula (or pendulums...) whose top ends extend a little bit above their axis, and are coupled to a pen suspended resting on a piece of paper placed on the platform of the Harmonograph.

The platform or the pendula are set in motion by hand. The oscillations decay, with the resulting curves getting smaller and smaller, spiralling in in a highly wobbly way. Weights are often located at various positions on the table to produce different oscillatory patterns, or the pendula lengths are adjusted.

The harmonograph was pioneered by the French physicist, Jules Antoine Lissajous in 1857. The first harmonograph actually used a light beam on a screen instead of the pens on paper that are used today. You can make your own by suspending a pencil flashlight from the ceiling by a number (e.g. 3) of strings (connect the flashlight by a few feet of string to the knot where you join the ceiling strings). Place a camera under the flashlight, darken the room and open the lens shutter for several seconds, and set the flashlight swinging in an arc.

Following the invention of the harmonograph it became a very popular device and was found in many homes. After the early 1900s it decreased in popularity and is rarely seen today.