Lissajous Ribbon
A 3:2 Lissajous figure drawn as a band that tapers to nothing at both ends.
About this piece
Drive the two axes of an oscilloscope with two sine waves and the spot traces a Lissajous figure:
x(θ) = sin(a·θ + δ) y(θ) = sin(b·θ)
When a and b are whole numbers the curve closes after one turn of θ, and the ratio decides the knot. This one is 3:2 — three lobes along the top, two down the side. Jules Antoine Lissajous produced the same shapes in 1857 by bouncing a light beam off small mirrors glued to two tuning forks at right angles.
It is not drawn as a hairline. At each of 420 samples the curve’s tangent is known exactly, because the derivative of a sine is a cosine: x′ = 3·cos(3θ+δ), y′ = 2·cos(2θ). Turn that 90° and you have the normal; push the centre line out along it by sin(πu), where u runs 0→1 across the sweep, and the band is zero-width at both ends and widest exactly halfway round. The outline is one closed polygon: up one edge, back along the other.
What to look for
The warm band is not a second figure. It is the same 3:2 curve running 0.86 radians behind in phase — a shadow of the shape, which is why it never wanders anywhere the cobalt band has not already been.
δ drifts at about a third of a radian per second, so the figure folds through itself and re-closes on a cycle of roughly twenty seconds. Watch the taper: once a turn, the ribbon narrows to a point and the piece is briefly held together by the hairline spine alone.
The two themes composite differently, on purpose. On the dark plate the bands are added, so crossings bloom; on the light paper they are laid down normally, because adding ink toward white is adding toward invisible.
Curious how the loop and canvas fit together? Read how it works →