What is a harmonograph?

A drawing machine in which swinging pendulums move a pen and a table, so the drawing records the combined motion.

Why do simple ratios make neat figures?

With a simple frequency ratio like 3:2 the pen comes back to nearly the same point after a few swings, so the curve closes up instead of wandering.

How long is a one-second pendulum?

About 25 cm for a full back-and-forth swing of one second; about 1 m for two seconds.

Can I save my drawing?

Yes. The download button saves a 1600 by 1600 pixel PNG print on paper.

Is a harmonograph the same as a spirograph?

No. A spirograph uses gears, so its curves repeat exactly. A harmonograph uses pendulums that slowly lose energy, so its curves spiral inward.

The Harmonograph

Two pendulums, one pen, and drawings nobody has made before. Back to the game

How to play

Two pendulums hang under a table. One swings the pen left and right, the other swings it forward and back. Set their lengths with the two sliders, press Swing again, and the pen traces the combined motion onto the card. Friction slowly takes the energy out of the swings, so every figure spirals inward and finishes itself after a minute or so. Change the pen colour between drawings to layer figures on top of each other, turn on Table spins for a third pendulum, and press the download button to save a large print of what you made.

Nice ratios jumps to lengths that make simple musical intervals. The star button sets a challenge: a faint drawing appears and you have to find the lengths that make it.

The physics: length sets the beat

A pendulum's period hardly depends on how heavy it is or how wide it swings. It depends on its length: T = 2π√(L/g), with g = 9.81 m/s². A pendulum 1 metre long takes about 2 seconds for a full swing; one 25 cm long takes about 1 second. To swing exactly twice as fast, a pendulum has to be a quarter of the length, which is why the sliders move in such uneven steps when you look for simple ratios.

The figure on the card is the sum of two damped oscillations, one along each direction. Mathematically each is a sine wave multiplied by a slowly shrinking exponential. When the two frequencies are in a simple ratio such as 3:2 the pen returns to almost the same place after a few cycles and the figure looks neat and closed. When the ratio is slightly off, the figure slowly rotates; when it is irrational, it never repeats at all.

Harmony you can see

The ratios that make the neatest figures are the same ones that sound good together in music. Pythagoras found that strings whose lengths are in the ratio 2:1 sound an octave apart and 3:2 a perfect fifth. The harmonograph turns those ratios into pictures: the HUD names the interval, and you hear the two frequencies played as notes when you move a slider. Close-but-not-quite ratios are the visual version of the slow wobble, or beating, you hear when two instruments are slightly out of tune.

A little history

Pendulum drawing machines became a craze in Victorian Britain. The Glasgow mathematician Hugh Blackburn is usually credited with an early version in 1844, and by the 1870s harmonographs were sold for parlours and science lectures. Their figures are relatives of Lissajous curves, which Jules Antoine Lissajous studied in 1857 using light reflected from vibrating tuning forks.

More art from physics: ../marble-paper/ and the singing sand of ../make-sand-sing/.

Questions people ask

What is a harmonograph?: A drawing machine in which swinging pendulums move a pen and a table, so the drawing records the combined motion.
Why do simple ratios make neat figures?: With a simple frequency ratio like 3:2 the pen comes back to nearly the same point after a few swings, so the curve closes up instead of wandering.
How long is a one-second pendulum?: About 25 cm for a full back-and-forth swing of one second; about 1 m for two seconds.
Can I save my drawing?: Yes. The download button saves a 1600 by 1600 pixel PNG print on paper.
Is a harmonograph the same as a spirograph?: No. A spirograph uses gears, so its curves repeat exactly. A harmonograph uses pendulums that slowly lose energy, so its curves spiral inward.