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 includ*e 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 build mathematical models for elaborate or unusual harmonogoraph machine configurations, in a search for new 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 paper table follows a** star-shaped** path instead of a

*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:*

**simple circle**Or imagine the paper table follows an even more arbitrary path, sort of a pretzel shape, and with concentric ripples out from the center, like this:

And consider one last example with a rotary paper table configuration. The four designs below are nearly identical from a mathematical perspective. The only difference is a slight detuning of oscillator frequencies which renders the designs very different to the eye. The lower right is with oscillators in phase to give straight lines, upper right is a classical harmonograph approach (slight detuning), and lower left is with the oscillators out of phase by π /4 to give ellipses.