What is a harmonograph?

A harmonograph is a mechanical device using two or more pendulums to draw geometric patterns on paper. Invented in the Victorian era (1840s–1890s), it works by attaching a pen to one pendulum and the paper to another, tracing compound Lissajous-like curves as both pendulums swing simultaneously. Damping causes the pattern to spiral inward over time.

What equations govern the harmonograph?

The harmonograph traces parametric curves described by x(t) = Ax·sin(fx·t + δx)·exp(−dx·t) and y(t) = Ay·sin(fy·t + δy)·exp(−dy·t), where fx/fy are the pendulum frequencies, δx/δy are phase offsets, dx/dy are exponential damping coefficients, and Ax/Ay are initial amplitudes. When the frequency ratio fx:fy is a simple integer ratio, closed Lissajous figures form before damping collapses the motion.

How does damping affect the harmonograph pattern?

Damping causes the oscillation amplitude to decay exponentially: A(t) = A₀·exp(−d·t). Higher damping collapses the pattern quickly, producing compact spiralling designs. Lower damping allows many cycles to complete before the pen reaches the centre. The differential damping of the x and y pendulums creates asymmetric spiral patterns. In real harmonographs, air resistance and bearing friction provide the damping.

What is the difference between a harmonograph and Lissajous curves?

Lissajous curves are undamped (d = 0), tracing perfectly closed loops that repeat indefinitely. A harmonograph adds exponential damping to both axes, so the pattern spirals inward toward the equilibrium point and eventually stops. Near-commensurate frequency ratios (like 2.01:3.00) produce slowly revolving Lissajous-like figures that never quite close — one of the most visually striking harmonograph patterns.

Home▸Drawing with Maths▸Harmonograph Simulator: Damped Pendulum Drawing Machine

🧪 Harmonograph Simulator: Damped Pendulum Drawing Machine

Interactive harmonograph simulator. Watch a damped compound pendulum trace intricate Lissajous-like curves. Adjust frequencies, phase, and damping to generate beautiful geometric art. Free browser-based physics simulation.

Drawing with Maths2DModerate60 FPS

⚙ Under the hood

This simulation recreates the Harmonograph, a device that produces complex drawings using a damped pendulum. It illustrates the relationship between mechanical motion and geometric patterns.

HarmonographDamped PendulumDrawing Machine

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install