Harmonographs #

A harmonograph is a mechanical apparatus that employs multiple coupled pendulums to create geometric images. As you can see below, the images are complex, yet beautiful.

These images can be created by attaching a pencil to the bottom of the pendulum. So, as the pendulum moves, these patterns are drawn.

A harmonograph on a single axis

A harmonograph creates its figures using the movements of damped pendulums. The movement of a damped pendulum along the $x-axis$ is described by the equation:

$$x(t) =Asin(2\pi ft+p)e^{dt}$$

where

• $A$ is the amplitude,
• $f$ is the frequency of the pendulum,
• $t$ is the time,
• $p$ is the phase,
• and $d$ is the damping factor.

If we have two independent pendulums oscillating along the x-axis, the principle ...