A Lissajous figure is a sound, measured rather than drawn

One continuous line, and it never lifts off the shirt. It loops back on itself again and again, crossing and re-crossing until it settles into a tight lattice, brighter wherever it crosses twice. Behind it, faint, a square grid — the trace of an oscilloscope screen with nothing else drawn on it.

Nothing about the shape was decided by an artist. It was measured.

What is a Lissajous figure actually a picture of?

Two sound frequencies plotted against each other instead of against time.

An ordinary waveform trace runs one signal along a horizontal timeline, amplitude against time. A Lissajous figure retires the clock altogether: one frequency drives the trace left and right, a second drives it up and down, and the dot draws whatever shape the two together make. Nothing about the drawing is invented — it's the direct, mechanical consequence of feeding an oscilloscope two tones at once and letting it plot one against the other, which is a different exercise from a single hit plotted against time on its own.

Why does the shape depend on the ratio between the two frequencies?

Because the ratio determines how many times each signal completes a cycle before the pattern repeats itself.

Two frequencies in a simple ratio, three cycles of one against two of the other, trace a figure that closes on itself quickly, into a shape with a handful of lobes. A more complicated ratio takes longer to close, or never quite closes at all, and produces a denser, busier lattice. The figure is a readout of that ratio the way a fraction is a readout of a division — change either frequency slightly and the whole shape reorganises itself around the new relationship.

Which ratio makes the most interesting figure to put on a shirt?

Whichever one the reader actually generated, rather than the simplest one on paper.

A one-to-one ratio draws a plain diagonal line or a circle, depending on the phase between the signals — accurate, but visually thin for a chest print. Ratios further from one-to-one draw denser lattices with more self-crossings, which is where the brightness the artwork is built around actually shows up. The interesting figure isn't the mathematically cleanest one, it's the one that came from two frequencies actually worth comparing — a note against a tuning reference, or, just as validly, a pulse against a fixed tempo.

Does anybody own the shape a Lissajous figure makes?

No — the figure is a mechanical consequence of two frequencies meeting, not a design anybody authored.

The same is true of the drawing style around it — an engraved trace on a grid, or a phosphor-green line on black, describes how it's rendered rather than who owns the underlying shape. What can be owned is a specific instrument's branding, its screen bezel, its control layout, none of which appears here. A figure generated by feeding two ordinary tones into an oscilloscope belongs to the mathematics, which was never anybody's to license in the first place.

Does this work for any two frequencies, or does it need to be musical?

Any two — the figure doesn't know or care whether either one is a musical note.

Two sine waves from a test bench draw exactly as valid a figure as two notes from an instrument tuned against a fixed reference. What changes the shape is the ratio, not the source, so a figure built from a heartbeat and a metronome is as legitimate as one built from a musical fifth. The design doesn't ask what the frequencies were, only what relationship they had.

Isn't a green line on a black grid just abstract decoration?

It reads as abstract only until somebody explains what generated it, and then it stops being decoration and starts being a specific answer.

An oscilloscope screen looks abstract to nobody who has used one — the grid and the green trace are the exact colours and shapes a scope actually displays, phosphor glowing on a dark screen. The figure itself isn't standing in for anything; it's the literal shape two frequencies make when compared, which makes it closer to a photograph of a relationship than to an illustration of one.

Does it print well?

Yes — a single continuous line on a near-black ground is about as forgiving as artwork gets.

The self-crossings want to stay genuinely brighter rather than just thicker — a slightly heavier line weight at each crossing reads as intensity without needing a second colour. One bright green against near-black keeps the whole figure legible at a distance, which matters here because the entire design is a single thin line rather than a filled shape.

How do I get one made?

Describe the two frequencies at JustOG — what they are, and roughly how they relate. Pick a direction, drag the crop frame, see it composited on the real garment, and it's made to order and shipped.

Designs other people have published are in the shop.

One line, never lifted, and the shape it settles into is the only thing anybody had to decide.