Two frequencies. One phase. The figure they trace together is never the same twice — unless the mathematics says it must be. Harmony made visible.
In 1857, French physicist Jules Antoine Lissajous attached a mirror to a vibrating tuning fork and directed a light beam at it. As two tuning forks vibrated at different frequencies, the combined motion of light traced complex, beautiful curves that captured the mathematical relationship between two sound waves.
Frequency ratios determine the shape completely. A 1:1 ratio makes an ellipse. A 1:2 ratio makes a figure-eight. Given the frequencies and phase offset, there is only one possible figure — deterministic and total.
Lissajous figures appear in physics laboratories, oscilloscopes, and acoustics. The same figure that describes a major third interval in music also describes a specific geometric shape. Sound and form are the same mathematics.
The classic Lissajous equations describe a 2D figure from two sine waves. The Szumski Tradition Engine extends this into three dimensions: a third frequency on the Z axis, then a 3D rotation before the figure is projected flat. The same deterministic mathematics — but now the curve exists in space, and the viewing angle reveals hidden structure.
A unique 3D Lissajous figure computed now. Three frequencies, three rotation axes, forty layered phase offsets. Every seed is a different figure — deterministic, reproducible, certifiable.
Your Lissajous figure was generated at this moment. The PDF records the exact frequency ratio and phase — a mathematically precise portrait of a sound relationship.
Get an email when new designs and curated collections go live. No marketing, just new work.