DIY · Card 56 · Physics · Art
A broom handle, a board on a Y of string and a felt-tip pen. Set the board swinging and the pen draws figures of eight, woven shells and spirals, the shapes of musical harmony, all by itself.
01The brief
Build a pendulum that draws pictures.
2 : 1
the knot a quarter of the way up: a figure of eight
3 : 2
the knot at 4⁄9: the musical fifth, drawn
1844
the year Hugh Blackburn described the Y-shaped pendulum
A pendulum swinging in a straight line would draw a straight line. But hang one from a Y-shaped string and it swings in two directions at once, at two different rhythms. Put paper on it and a pen above it, and those two rhythms draw themselves: figures of eight, looping knots and slowly shrinking shells.
This is a harmonograph, a Victorian marvel you can make from a broom handle and some string. Move one paperclip and you change the drawing completely. It is a close relative of the pendulum wave, which also plays rhythms against each other, and of the moiré patterns made by two almost-matching grids.
02Kit
String, a board and a pen. The chairs do the heavy lifting.
Tick off what you have
0 of 6 ready
03Method
The paperclip on the Y is the tuning knob. Everything else just needs to be steady.
Step 1 of 8 · Minute 0
You need a broom handle or long dowel, two tall chairs, about 4 metres of strong string, a key ring and a paperclip, a stiff board about 30 cm square, a few heavy books, sheets of paper, a felt-tip pen, a strip of stiff card, tape and a ruler.
The drawing is a smudge of thick lines.
Why
The pen is pressing too hard and dragging the board.
Fix
Let the pen rest on the paper by its own weight, with the card hinge free to rise and fall.
The swing dies away in seconds.
Why
The board is too light, or the bar is wobbling.
Fix
Add more books, tape the bar down firmly and use a smooth, fine felt-tip.
It just draws a line or an oval.
Why
The board was swung straight across, or the knot is at the top.
Fix
Pull the board out diagonally, so both swings start, and move the paperclip down the Y.
04Two rhythms
A pendulum’s rhythm depends only on its length. Swinging across the Y, the whole length of string, from the bar down to the board, swings as one: a long, slow pendulum. Swinging along the Y, the two arms stay put and only the string below the knot moves: a shorter, faster one. Swing the board diagonally and it does both at once.
Slide the knot and see which ratio you get:
Frequency ratio
2.000 : 1
Nearest interval
2:1 octave
For a 2:1 figure, put the knot so that ℓ is 25.0 % of L. A 2:1 figure of eight needs the knot a quarter of the way up from the board; a 3:2 needs it at 4⁄9, about 44 %. Anywhere in between, the ratio is not a simple fraction and the figure never quite closes.
The pen adds the two swings together, one sideways, one up and down, while friction slowly shrinks them both. Choose a ratio and draw:
A 3:2 ratio, the musical perfect fifth: one swing makes 3 trips across while the other makes 2. Exactly in tune, the pen retraces one shape as it shrinks. A fraction out of tune, the shape slowly turns as it goes, weaving the woven, shell-like figures harmonographs are famous for. Friction sets how fast it spirals in.
05The big idea
Harmony
you can see.
Two notes sound sweet together when their frequencies make a simple ratio. Two swings draw a neat figure for exactly the same reason: every few swings, they come back into step.
The harmonograph turns that hidden arithmetic into pictures. A 2 : 1 octave is a figure of eight; a 3 : 2 fifth, a pretzel; a ratio that is nearly but not quite simple, a figure that slowly turns. Your eye can hear harmony.
06The math
1 · The figure
x = e^(−kt) sin(f₁t + φ), y = e^(−kt) sin(f₂t)
Two swings at frequencies f₁ and f₂, dying away at rate k.
2 · A pendulum’s rhythm
f ∝ 1 ⁄ √L
Frequency goes as one over the square root of length.
3 · The Y
f_along ⁄ f_across = √(L ⁄ ℓ)
Full length L across, length ℓ below the knot along.
For a 2 : 1 ratio, √(L ⁄ ℓ) = 2, so ℓ = L ⁄ 4. With the board 1 m below the bar, the knot goes 25 cm above the board. For 3 : 2, ℓ = 4L ⁄ 9 ≈ 44 cm; for 3 : 1, ℓ = L ⁄ 9 ≈ 11 cm.
The across swing of a 1 m pendulum takes about 2 s; the along swing at 25 cm about 1 s. A ratio that is out by just 1 % makes the figure turn once every hundred swings, which is why a small nudge of the clip makes such a difference.
07Real life
From Victorian drawing rooms to oscilloscope screens.
Oscilloscopes
Feed two signals into an oscilloscope, one moving the dot sideways and one up and down, and it draws the same figures. Engineers once used them to compare two frequencies: a still figure of eight means exactly 2 : 1.
Musical intervals
An octave is two notes whose frequencies are in the ratio 2 : 1; a perfect fifth, 3 : 2; a fourth, 4 : 3. These simple ratios sound sweet together, and they draw the neatest harmonograph figures.
Tuning forks and mirrors
In 1857 Jules Antoine Lissajous fixed small mirrors to two tuning forks at right angles and bounced a beam of light off both onto a screen. The vibrating forks traced glowing figures, now called Lissajous curves.
The ABC logo
The logo of the Australian Broadcasting Corporation, introduced in 1965, is a Lissajous figure, the shape an oscilloscope draws with signals in a 3 : 1 ratio.
Victorian parlours
In the late 1800s harmonographs became a fashionable drawing-room novelty. Guests set the pendulums swinging and took home the drawings, and books of the most beautiful figures were published.
Sand pendulums
Hang a leaking bottle of sand from a Y-shaped string, the arrangement Hugh Blackburn described in 1844, and it pours harmonograph figures onto the floor. Science museums still use the idea.
1815
The American mathematician Nathaniel Bowditch, famous for his book on navigation, studied these curves in 1815 with a pendulum hung from two points. In 1844 Hugh Blackburn, a professor of mathematics in Glasgow, described the Y-shaped version you have just built, and in 1857 the French physicist Jules Antoine Lissajous made the same figures dance in light with tuning forks.
So the curves have three names: Bowditch curves, Blackburn figures and, most often, Lissajous figures. By the end of the century, harmonographs had become a craze in Victorian homes.
08Is and isn’t
The patterns are random.
They are completely determined by the two rhythms, where you start the swing and how quickly it dies away. Set up the same swing twice and you get the same drawing.
You need a complicated machine.
A single board on a Y-shaped string, a broom handle and a pen is enough to draw the classic figures. Elaborate harmonographs simply add more pendulums.
Perfectly tuned is always best.
Exactly in tune, the pen retraces the same shape. A slight mistuning makes the figure turn slowly as it shrinks, which gives the most beautiful woven drawings.
A heavier board swings faster.
Weight doesn’t change a pendulum’s rhythm, only its length does. Extra weight just keeps it swinging longer, so the drawing has more lines.
09Go further
A
A gallery of ratios
Draw a figure for each knot position: a quarter, 4⁄9, 9⁄16 and 1⁄9. Label each with its ratio and musical interval.
B
Just out of tune
Set the knot exactly for 2 : 1, then move it a centimetre. How does the figure change?
C
Colour layers
Draw one figure in blue, then swing again from a slightly different start and draw over it in green.
D
A sand pendulum
Hang a plastic cup with a pinhole, filled with salt, from the same Y over dark paper outdoors.
E
Count the loops
In a still, in-tune figure, count the loops touching the top edge and the side edge. What is their ratio?
10Questions
A machine that uses swinging pendulums to draw. Two or more swings, at different rhythms, move a pen and a sheet of paper relative to each other, and as the swings slowly die away the pen draws looping, shrinking geometric figures. It became a popular Victorian scientific toy.
Hang a weighted board from a broom handle on a Y-shaped string, the two arms of the Y joined by a sliding paperclip. Tape paper to the board and hold a pen lightly on it from a fixed arm. Swing the board diagonally and the pen draws a figure that depends on where the clip sits.
Swinging across the Y, the whole string pivots from the bar, so the pendulum is long and slow. Swinging along the Y, the arms stay fixed and only the string below the knot swings, so it is shorter and faster. The ratio of the two frequencies is the square root of the full length divided by the length below the knot.
It can’t be credited to one person. Nathaniel Bowditch studied these curves with a pendulum hung from two points in 1815, and Hugh Blackburn, a mathematics professor in Glasgow, described a Y-shaped pendulum in 1844. Jules Antoine Lissajous made the same figures with tuning forks and mirrors in 1857.
The pleasing figures come from simple frequency ratios, 2 : 1, 3 : 2, 4 : 3, and these are the same ratios as the musical octave, fifth and fourth. A harmonograph lets you see the intervals that sound harmonious, which is how it got its name.
Safety · Tape the bar and books down firmly so nothing heavy can fall, and keep small children clear of the swinging board.
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