Harmonograph from spherical pendulums

Spherical pendulums produce some attractive forms on their own, but they have been applied together for a long time now (it is quite a Victorian thing apparently) to create compound shapes. The machines that make these shapes are called harmonographs, so called due to their use in helping to visualise musical harmonies. As far as I know, Harmonographs have only ever been curiosities, without any real practical application beyond the production of attractive postcards.

Harmonographs use pendulums in a variety of configurations to produce different shapes, dragging a pen over paper to produce a shape. Some use simple pendulums in combination with spherical pendulums. In every case, the complexity and beauty of the form produced is much greater than the crude contraption it has come from.

A few harmonographs have been produced using light and long exposure and produce very beautiful images:

One artist (artist collective) has already done some of this before, though without interaction between the physical and the digital. I hear they are a big deal, so very happy to follow in their footsteps.

Someone online has made a 3D harmonograph using python (as I hope to do very soon!) but has just used sinusoidal patterns rather than any proper decay functions to try to model a spherical pendulum – I am not even sure they have used a model of a spherical pendulum rather than a series of simple pendulums. I wonder how mine might be different.

Solution for a simple harmonograph’s motion is here:

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