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.

Harmonograph Simulator

Watch a virtual damped compound pendulum draw intricate parametric curves. Adjust frequency ratios, phase offsets, and damping to generate spiralling geometric art — the mathematics of Victorian drawing machines.

FREE Physics Mathematics NEW · Session 44

Ratio 2:3

Time 0.0s

Points 0

Amplitude 100%

Playback

Presets (1–6)

Frequency

X Frequency 2.00

Y Frequency 3.00

Phase Offset

Phase X 0.00

Phase Y 1.57

Damping

Damp X 0.0020

Damp Y 0.0020

Amplitude

Amp X 1.00

Amp Y 1.00

Speed

Draw Speed 0.40

Display

Rainbow trail Persist layers

⌨ Keyboard Shortcuts

Space Play / Pause

C Clear canvas

S Sweep phase toggle

1 Classic preset

2 Figure-8 preset

3 Infinity preset

4 Web preset

5 Butterfly preset

6 Rose preset

What is a Harmonograph?

A harmonograph is a mechanical device that uses swinging pendulums to draw geometric patterns. First popularised in the Victorian era (1840s–1890s), it typically consists of two or three pendulums: one moves a pen in the x-direction, another moves paper in the y-direction, and an optional third adds rotational motion. As the pendulums swing simultaneously, the pen traces intricate Lissajous-like curves on paper.

The word "harmonograph" comes from the Latin harmonia (agreement, consonance) and Greek grapho (I write). Victorian scientists including Hugh Blackburn and Jules Antoine Lissajous studied how frequency ratios between the pendulums determine the family of curves produced — a deep connection between music, mathematics, and visual art.

Victorian Invention

Popularised in the 1840s–1890s. Exhibited at the Great Exhibition (1851) and became a fashionable parlour scientific instrument in Victorian homes and laboratories.

Damped Oscillation

Real pendulums lose energy to air resistance and friction, causing the amplitude to decay exponentially: A(t) = A₀·e^−dt. This damping creates spiralling inward patterns.

Frequency Ratios

Simple integer ratios (2:3, 3:4) produce clean Lissajous figures. Near-integer ratios (2.01:3.00) create slowly precessing patterns that reveal the underlying Lissajous structure.

Phase Sensitivity

The phase offset δ between the x and y pendulums determines the "rotation" of the Lissajous figure. At δ = 0 the figure is a line, at δ = π/2 an ellipse, at δ = π/4 a tilted figure.

The Mathematics of the Harmonograph

This simulation computes the position of the pen at each time step using the damped simple harmonic oscillator equations for each axis:

Parametric Equations

x(t) = A_x · sin(f_x·t + δ_x) · e^−d_x·t

y(t) = A_y · sin(f_y·t + δ_y) · e^−d_y·t

Where f_x, f_y are angular frequencies, δ_x, δ_y are phase offsets, d_x, d_y are damping coefficients, and A_x, A_y are initial amplitudes.

Each equation describes a damped simple harmonic oscillator — the same mathematics that governs a mass-on-a-spring with friction, an LC circuit with resistance, or a pendulum with air drag. The product of the sinusoidal oscillation and the exponential decay envelope produces oscillation that diminishes over time.

Frequency Ratio Table

Ratio f_x:f_y	Pattern Family	Musical Interval	Symmetry
1:1	Ellipse / Circle	Unison	Infinite rotational
1:2	Figure-8 / Parabola	Octave	Bilateral
2:3	Classic Lissajous bow	Perfect Fifth	2-fold rotational
3:4	Three-loop figure	Perfect Fourth	3-fold rotational
3:5	Five-petal rose	Minor Sixth	3-fold rotational
Near-integer	Slowly rotating pattern	Slightly detuned	None (quasi-periodic)

Damping and Energy Dissipation

The exponential decay term e^−dt models energy dissipation. For a physical pendulum, d = b/(2m) where b is the linear damping coefficient and m is the mass. The time constant τ = 1/d gives the time for amplitude to fall to 1/e ≈ 37% of its initial value.

Asymmetric damping (d_x ≠ d_y) causes one axis to collapse faster than the other, producing patterns that morph from 2D Lissajous figures toward 1D oscillations as time progresses — a distinctly harmonograph phenomenon not seen in undamped Lissajous curves.

Lissajous Curves vs Harmonograph Patterns

Lissajous curves (studied by Jules Antoine Lissajous in 1857) are the undamped special case d = 0. They produce perfectly closed, time-independent curves that serve as the geometric skeleton of harmonograph patterns. When d → 0 in this simulator, the spiralling slows and the pattern approaches a stable Lissajous figure.

Lissajous figures are widely used in electronics and physics: on an oscilloscope, connecting two sinusoidal signals to the X and Y channels produces Lissajous figures whose shape reveals the frequency ratio and phase relationship between the signals. This makes them essential for calibrating audio equipment, testing electronic filters, and measuring phase angles in AC circuits.

The Phase Sweep Effect

Click "Sweep Phase" (or press S) to continuously animate the phase offset δ_y. As δ sweeps from 0 to 2π, the Lissajous figure rotates through all orientations — from a tilted line (δ=0,π) through ellipses to a circle (δ=π/2). This is exactly what oscilloscopes display when one signal is slightly off-frequency from another, producing the characteristic "breathing" Lissajous pattern used in frequency comparisons.

Curriculum Connections

Level	Subject	Topics Covered
GCSE	Physics	Waves, oscillations, energy dissipation
A-Level	Physics	Simple harmonic motion, damping, resonance
A-Level Further Maths	Mathematics	Parametric equations, differential equations, exponential decay
IB Physics HL	Physics	Oscillations and waves: SHM, damping, phase
IB Maths AA HL	Mathematics	Parametric functions, complex oscillations
University Year 1	Physics/Engineering	Coupled oscillators, Fourier analysis, damped SHM