The design space is nearly infinite for harmonographs, when the definition is somewhat broadened. The design space must of course include classical harmonograph machines, based on 2-3 pendulums, that are driven only by gravity and subject to friction or damping. The design space should also include pintograph machines, or machines based on multiple rotating disks that are connected together through linkages. And further, the design space might also include other more complicated harmonograph machines that have not yet been built, including machines that are too elaborate for construction. These machines might employ strange combinations of sinusoidal and non-sinusoidal oscillators, or weird damping profiles, or unusual paper table paths, or require external power through motors or hand cranks in order to overcome damping (friction).
My design approach is to mathematically describe elaborate or non-physical machine configurations in a search for new harmonograph design forms.
As a starting example, the graphic below shows how a classic 2-3 pendulum design may be transformed into a completely different pattern by a rotational sweep or linear translation, in other words, by adding only a rotary or linear paper table.
As the next step, imagine that the rotary paper motion follows a star-shaped path instead of a simple circle and that the elliptical oscillations are collapsed to nearly straight lines (ala linearly polarized light). These two modifications would result in something more interesting than a classic 2-3 pendulum design, like this:
Star-shaped paper path with straight-line oscillator
And consider one last example, also with a rotary paper table configuration. The four designs below are nearly identical from a mathematical perspective. The only difference is a very slight detuning of oscillator frequency – and yet the designs appear quite different to the human eye and brain. The upper right design is a classical harmonograph approach. The lower left design looks like a planetary system and might be actually be produced with planetary gears.