|For a quick start, check the Color button, then click the Random button a few times.|
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?
|Encyclogram draws harmonographs, 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
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).
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.
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 -.
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.
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.
Can be downloaded as an application on download page