Harmonograph with decay
A harmonograph is a mechanical drawing device where a pen is suspended from two pendulums swinging in perpendicular directions. As friction steals energy, the pendulums decay and the pen traces spiraling, evolving curves — not quite Lissajous figures, because the amplitude is always shrinking. The result is somewhere between mathematics and art.
x(t) = A1e−d1t sin(ω1t + φ1) | y(t) = A2e−d2t sin(ω2t + φ2)
The harmonograph
The harmonograph was a popular drawing machine in Victorian parlors, first described around 1844. A pen is mounted on a platform connected to two (or more) pendulums swinging in perpendicular directions. As the pendulums swing, the pen traces a compound curve on paper. The device fascinated Victorians because it produced mathematical beauty from simple mechanics — no artistic skill required.
Frequency ratios and musical intervals
The character of the pattern depends on the ratio of the two pendulum frequencies. Simple ratios produce organized patterns: 1:1 gives ellipses (unison), 2:3 gives trefoil shapes (a perfect fifth in music), 3:4 gives four-lobed roses (a perfect fourth). Irrational ratios produce patterns that never close, filling the plane densely — the visual equivalent of dissonance.
The role of decay
Without decay, the harmonograph traces a perfect Lissajous figure that repeats forever. Friction changes everything. As the amplitudes shrink exponentially, the pattern spirals inward, creating layers of nested curves that get progressively tighter. The decay rate controls how quickly this happens: slow decay produces many layers; fast decay gives a loose, open spiral. Different decay rates on each axis create asymmetric patterns.
The mathematics
Each axis follows a damped sinusoidal motion: x(t) = A·e−dt·sin(ωt + φ). The full pattern is a parametric curve in (x, y) parameterized by time. With rational frequency ratios, the underlying Lissajous figure is periodic with period 2π / gcd(ω1, ω2). The decay envelope e−dt breaks this periodicity, giving each pass through the pattern slightly smaller amplitude than the last.
Phase and symmetry
The phase offset φ between the two pendulums controls the pattern’s orientation and symmetry. At φ = 0, the pattern is symmetric about the diagonal. At φ = π/2, it’s symmetric about the axes. Intermediate values rotate the figure and break symmetries, producing the most complex and visually interesting patterns.