Interactive Lab

3D Harmonograph

Three damped pendulums control the x, y, and z coordinates of a pen tracing a path through three-dimensional space. The resulting curve is a damped 3D Lissajous figure — when the frequency ratios are integers, the path closes into elegant knotted loops before spiraling inward as energy dissipates. Drag to rotate the view; adjust frequency ratios, damping, and phase offsets to explore the space of possible curves.

x(t) = A e^{−d t} sin(f_x t + φ_x), y(t) = A e^{−d t} sin(f_y t + φ_y), z(t) = A e^{−d t} sin(f_z t + φ_z)

Points: 0

Ratios: 2:3:4

Drag to rotate

Freq X 2

Freq Y 3

Freq Z 4

Phase X 0

Phase Y 0

Phase Z 0

Damping 0.002

Trail Length 2000

Speed 1.0

Presets

From 2D to 3D

A traditional harmonograph uses two pendulums to trace a 2D curve. By adding a third pendulum controlling the z-axis, the curve lifts off the plane and traces a path through three-dimensional space. The result is related to 3D Lissajous curves, but with exponential damping that causes the pattern to spiral inward over time, creating layered structures that would be impossible with undamped oscillators.

Frequency Ratios and Musical Intervals

When the frequency ratios are simple integers, the curve closes (or nearly closes) into repeating patterns. The ratio 1:2 corresponds to an octave, 2:3 to a perfect fifth, 3:4 to a perfect fourth. Just as consonant musical intervals produce pleasing sounds, consonant frequency ratios produce aesthetically pleasing visual patterns. Irrational ratios produce curves that never exactly repeat, eventually filling a volume.

The Mathematics

Each coordinate is governed by a damped sinusoid: x(t) = A e^-dt sin(f_xt + φ_x). The amplitude A sets the initial size, d controls the damping rate (energy loss per cycle), f sets the frequency, and φ the initial phase. The phase offsets are crucial — even with the same frequency ratios, different phases produce dramatically different curves. The total parameter space is six-dimensional (three frequencies, three phases), plus damping and amplitude.

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