A harmonograph is a mechanical device using two or more pendulums to guide a pen across paper, producing intricate Lissajous-like curves. The motion of each pendulum is described by a damped sinusoid: x(t) = A sin(f₁t + φ₁)·e^(−d₁t) and y(t) = B sin(f₂t + φ₂)·e^(−d₂t), where f is frequency, φ is phase offset, and d is the damping coefficient. When the frequency ratio f₁/f₂ is a simple rational number (such as 1:1, 2:3, or 3:4) the curve closes into a recognisable Lissajous figure; irrational ratios produce open curves that gradually spiral inward as damping reduces the amplitude, creating the characteristic dense spirograph-like shapes seen in Victorian science demonstrations.
This simulator lets you tune each pendulum's frequency, phase, amplitude, and damping independently. Explore how small changes to the frequency ratio shift closed forms to open spirals, and how increasing damping collapses the figure faster.
What is the difference between a harmonograph and a Lissajous figure?
A Lissajous figure is the undamped, idealised case: x = A sin(f₁t + φ), y = B sin(f₂t). It produces a perfectly closed, stable curve that repeats indefinitely. A harmonograph adds exponential damping — e^(−dt) — so the amplitude shrinks over time and the curve spirals inward, never perfectly repeating. The harmonograph curve converges to the origin if both pendulums damp; otherwise it converges to a smaller closed Lissajous figure.
When does a harmonograph curve close?
In the absence of damping, the curve closes if and only if the frequency ratio f₁/f₂ is a rational number p/q (in lowest terms). The curve makes q complete cycles of x and p complete cycles of y before returning to the start, and a closed figure has exactly 2(p − 1) or 2(q − 1) self-crossings depending on orientation. With damping the curve technically never closes — it spirals inward — but nearly-rational ratios produce patterns that appear to close before the amplitude becomes negligible.
What role does phase play in harmonograph patterns?
The phase difference φ₁ − φ₂ controls the orientation and shape of the figure. At φ = 0 or π a 1:1 ratio gives a straight diagonal line; at φ = π/2 it gives a circle (if amplitudes match). For a 2:3 ratio, varying the phase continuously rotates the three-lobed figure through its family of orientations. In physical harmonographs the phase is set by releasing the pendulums at different points in their swing, making it the most sensitive parameter to tune.
Lissajous figures are used to measure the frequency ratio and phase difference between two sinusoidal signals using an oscilloscope in XY mode. If the figure is a closed ellipse, the phase difference φ can be read from the ratio of the y-intercept to the y-amplitude: sin(φ) = y₀/yₘₐₓ. Counting the number of loops along each axis gives the frequency ratio directly: n_x tangencies on the left edge and n_y tangencies on the top edge indicate the ratio n_x : n_y.
A typical lateral harmonograph has two pendulums whose motion is coupled through a pen-and-paper arrangement: one pendulum drives x-deflection, the other drives y-deflection, and the pen traces the combined path. Some designs use a third "rotating" pendulum to add a rotational component. The device was popularised in Victorian England; the Crystal Palace Exhibition of 1851 featured one, and it became a fashionable scientific toy used to demonstrate wave mathematics to lay audiences.
In a physical harmonograph the damping constant d depends on air resistance and pivot friction. For a simple pendulum in air, d ≈ b/(2m) where b is the linear drag coefficient and m is the pendulum mass. Longer, heavier pendulums have smaller d — they damp more slowly — which is why harmonograph designers use large brass pendulum bobs. In this simulation you can set d independently for each axis, allowing asymmetric damping that creates asymmetric spiralling patterns not achievable in a simple physical device.
When f₁ ≈ f₂ but not exactly equal, the harmonograph exhibits a beat pattern: the phase difference between the two pendulums slowly drifts, causing the figure to rotate gradually through its family of orientations. The beat frequency is |f₁ − f₂|, and the period of one full rotation of the figure is 1/|f₁ − f₂|. This is the same phenomenon as acoustic beating between two nearly-tuned instruments, but visualised geometrically as a slowly rotating Lissajous figure.
No — harmonograph curves are restricted to the family of parametric curves of the form (A sin(f₁t + φ₁)·e^(−d₁t), B sin(f₂t + φ₂)·e^(−d₂t)). Adding more pendulums (up to four is common) allows more complex Fourier-like superpositions, but a finite harmonograph can only produce finite Fourier sums. Arbitrary closed curves require infinitely many frequency components, as shown by the Fourier series — this is the principle behind the Fourier Epicycles simulator.
A frequency ratio of 1:2 with a phase difference of π/2 produces a figure-8 (lemniscate-like shape). The x-motion completes one cycle while y completes two, creating two loops. At phase difference 0 or π the same 1:2 ratio produces a parabola. The exact shape and orientation depend sensitively on the phase, making the 1:2 Lissajous family one of the richest for visual exploration.