Success! I looked through this guy's site and found the equations for the x-y motion of the "pen" drawing a harmonograph. He shows a Mathematica program that uses sliders to dynamically change the frequencies, phase shifts and decay constants and immediately see the finished harmonograph.
When I wrote those equations in a Python program it came out perfect. It takes my program a while to draw the image, but then so do real harmonographs.
I had other ideas, though: moving the sliders, I couldn't help but feel like the images were a 3D object being rotated. Then it hit me: I could draw harmonographs using a Visual Python program where one sphere (the "pen") goes back and forth in all 3 planes! I think it worked pretty well. Check it out:
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