Encyclogram: Harmonographs, Spirographs, & Lissajous
|Click here for a larger encyclogram: 800x600, 1024x768|
|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.|
A harmonograph is a mechanical apparatus that employs pendulums to create a geometric image. The drawings created typically are Lissajous curves, or related drawings of greater complexity. The devices, which began to appear in the mid-19th century and peaked in popularity in the 1890's, cannot be conclusively attributed to a single person, although Hugh Blackburn, a professor of mathematics at the University of Glasgow, is commonly believed to be the official inventor.
A simple harmonograph uses two pendulums to control the movement of a pen relative to a drawing surface. One pendulum moves the pen back and forth along one axis and the other pendulum moves the drawing surface back and forth along a perpendicular axis. By varying the frequency of the pendulums relative to one another (and phase) different patterns are created. Even a simple harmonograph as described can create ellipses, spirals, figure eights and other Lissajous figures.
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.
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.
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* 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.
If you have Logo, try these:
cs make "dk 200 repeat 21600 [setxy (sin(2 * repcount)) * :dk (sin(3 * repcount)) * :dk make "dk :dk * .9999] ht
cs make "dk 300 repeat 60 [repeat 360 [setpc repcount/22.5 setxy (sin(2 * repcount)) * :dk (sin(3 * repcount)) * :dk make "dk :dk * .9999]] ht
Try changing the 2 and 3 (* repcount) to other numbers.
A graph is a two-dimensional representation of data. A typical example would be a graph of some quantity varying with time, e.g. daily temperature. In science, we often use graphs to give us a picture of the relationships between variables.