Sand pendulum
A pendulum swings freely in two axes while releasing a thin stream of sand onto a tray below. As it traces compound curves, the sand accumulates — brighter where the path overlaps itself repeatedly. The result is a physical Lissajous figure drawn in golden sand, slowly decaying toward stillness.
x(t) = A e−dt sin(ωxt) | y(t) = A e−dt sin(ωyt + φ)
The sand pendulum
A sand pendulum is a physical device where a funnel of sand hangs from a two-axis pivot point — typically a gimbal or Y-frame that allows the pendulum to swing freely in both the x and y directions simultaneously. As it swings, sand trickles through a small hole in the funnel, leaving a visible trace on a dark surface below. The device has been used in science museums and classrooms for decades as a tangible demonstration of harmonic motion and Lissajous curves.
Lissajous figures in sand
The pattern created depends on the ratio of frequencies in the two swing directions. If the pendulum swings twice as fast in x as in y, the ratio is 1:2, producing a figure-eight. A 2:3 ratio produces a trefoil. Simple integer ratios create clean, closed figures; irrational ratios produce patterns that never exactly repeat, gradually filling the tray. The phase offset — the relative timing of the two swings — rotates and reshapes the figure.
Why sand accumulates
As the pendulum’s amplitude decays due to friction and air resistance, each successive pass through the pattern has slightly smaller amplitude than the last. The sand trace thickens where the pendulum spends the most time — near the turning points of each swing, where the bob momentarily slows. In this simulation, pixel brightness represents sand depth: brighter gold means more grains have been deposited at that location.
The physics of decay
Real pendulums lose energy to air drag (proportional to velocity) and pivot friction. The amplitude follows approximately exponential decay: A(t) = A0 e−γt. Different damping rates in x and y (common in physical setups due to asymmetric mounts) create asymmetric patterns that evolve as the pendulum runs down. The interplay between the frequency ratio, phase, and decay is what makes each sand pendulum run unique.
From device to art
Sand pendulums are closely related to harmonographs, Victorian-era drawing machines that use coupled pendulums to move a pen. Both devices create parametric curves governed by damped sinusoidal motion. The sand version has a special aesthetic quality: the granular medium creates a textured, luminous trace that builds up over time, making visible the accumulation of motion.