## The Harmonograph

Anita Chowdry is an artist based in London. While many are exploring electronic media and computers, she’s going in the opposite direction, exploring craftsmanship and the hands-on manipulation of matter. I find this exciting, perhaps because I spend most of my days working on my laptop, becoming starved for richer sensations. She writes:

Today, saturated as we are with the ephemeral intangibility of virtual objects and digital functions, there is a resurgence of interest in the ingenious mechanical contraptions of pre-digital eras, and in the processes of handcraftsmanship and engagement with materials. The solid corporality of analogue machines, the perceivable workings of their kinetic energy, and their direct invitation to experience their science through hands-on interaction brings us back in touch with our humanity.

The ‘steampunk’ movement is one way people are expressing this renewed interest, but Anita Chowdry goes a bit deeper than some of that. For starters, she’s studied all sorts of delightful old-fashioned crafts, like silverpoint, a style of drawing used before the invention of graphite pencils. The tool is just a piece of silver wire mounted on a writing implement; a bit of silver rubs off and creates a gray line. The effect is very subtle:

In January she went to Cairo and worked with a master calligrapher, Ahmed Fares, to recreate the title page of a 16th-century copy of Avicenna’s Canon of Medicine, or al-Qanun fi’l Tibb:

This required making gold ink:

The secret is actually pure hard work; rubbing it by hand with honey for hours on end to break up the particles of gold into the finest powder, and then washing it thoroughly in distilled water to remove all impurities.

The results:

I met her in Oxford this March, and we visited the Museum of the History of Science together. This was a perfect place, because it’s right next to the famous Bodleian, and it’s full of astrolabes, sextants, ancient slide rules and the like…

… and one of Anita Chowdry’s new projects involves another piece of romantic old technology: the harmonograph!

### The harmonograph

A harmonograph is a mechanical apparatus that uses pendulums to draw a geometric image. The simplest so-called ‘lateral’ or ‘rectilinear’ harmonograph uses two pendulums: one moves a pen back and forth along one axis, while the other moves the drawing surface back and forth along a perpendicular axis. By varying their amplitudes, frequencies and the phase difference, we can get quite a number of different patterns. In the linear approximation where the pendulums don’t swing too high, we get Lissajous curves:

$x(t) = A \sin(a t + \delta)$

$y(t) = B \sin(b t)$

For example, when the amplitudes $A$ and $B$ are both 1, the frequencies are $a = 3$ and $b = 4,$ and the phase difference $\delta$ is $\pi/2,$ we get this:

Harmonographs don’t serve any concrete practical purpose that I know; they’re a diversion, an educational device, or a form of art for art’s sake. They go back to the mid-1840s.

It’s not clear who invented the harmonograph. People often credit Hugh Blackburn, a professor of mathematics at the University of Glasgow who was a friend of the famous physicist Kelvin. He is indeed known for studying a pendulum hanging on a V-shaped string, in 1844. This is now called the Blackburn pendulum. But it’s not used in any harmonograph I know about.

On the other hand, Anita Chowdry has a book called The Harmonograph. Illustrated by Designs actually Drawn by the Machine, written in 1893 by one H. Irwine Whitty. This book says the harmonograph

was first constructed by Mr. Tisley, of the firm Tisley and Spiller, the well-known opticians…

So, it remains mysterious.

The harmonograph peaked in popularity in the 1890s. I have no idea how popular it ever was; it seems a rather cerebral form of entertainment. As the figures from Whitty’s book show, it was sometimes used to illustrate the Pythagorean theory of chords as frequency ratios. Indeed, this explains the name ‘harmomograph’:

At left the frequencies are exactly $a = 3, b = 2,$ just as we’d have in two notes making a major fifth. Three choices of phase difference are shown. In the pictures at right, actually drawn by the machine, the frequencies aren’t perfectly tuned, so we get more complicated Lissajous curves.

How big was the harmonograph craze, and how long did it last? It’s hard for me to tell, but this book published in 1918 gives some clues:

• Archibald Williams, Things to Make: Home-made harmonographs (part 1, part 2, part 3), Thomas Nelson and Sons, Ltd., 1918.

It discusses the lateral harmonograph. Then it treats Joseph Goold’s ‘twin elliptic pendulum harmonograph’, which has a pendulum free to swing in all directions connected to a pen, and second pendulum free to swing in all directions affecting the motion of the paper. It also shows a miniature version of the same thing, and how to build it yourself. It explains the connection with harmony theory. And it explains the value of the harmonograph:

#### Value of the harmonograph

A small portable harmonograph will be found to be a good means of entertaining friends at home or elsewhere. The gradual growth of the figure, as the card moves to and fro under the pen, will arouse the interest of the least scientifically inclined person; in fact, the trouble is rather to persuade spectators that they have had enough than to attract their attention. The cards on which designs have been drawn are in great request, so that the pleasure of the entertainment does not end with the mere exhibition. An album filled with picked designs, showing different harmonies and executed in inks of various colours, is a formidable rival to the choicest results of the amateur photographer’s skill.

“In great request”—this makes it sound like harmonographs were all the rage! On the other hand, I smell a whiff of desperate salesmanship, and he begins the chapter by saying:

Have you ever heard of the harmonograph? If not, or if at the most you have very hazy ideas as to what it is, let me explain.

So even at its height of popularity, I doubt most people knew what a harmonograph was. And as time passed, more peppy diversions came along and pushed it aside. The phonograph, for example, began to catch on in the 1890s. But the harmonograph never completely disappeared. If you look on YouTube, you’ll find quite a number.

### The harmonograph project

Anita Chowdry got an M.A. from Central Saint Martin’s college of Art and Design. That’s located near St. Pancras Station in London.

She built a harmonograph as part of her course work, and it worked well, but she wanted to make a more elegant, polished version. Influenced by the Victorian engineering of St. Pancras Station, she decided that “steel would be the material of choice.”

So, starting in 2013, she began designing a steel harmonograph with the help of her tutor Eleanor Crook and the engineering metalwork technician Ricky Lee Brawn.

Artist and technician David Stewart helped her make the steel parts. Learning to work with steel was a key part of this art project:

The first stage of making the steel harmonograph was to cut out and prepare all the structural components. In a sense, the process is a bit like tailoring—you measure and cut out all the pieces, and then put them together an a logical order, investing each stage with as much care and craftsmanship as you can muster. For the flat steel components I had medium-density fibreboard forms cut on the college numerical control machine, which David Stewart used as patterns to plasma-cut the shapes out of mild carbon-steel. We had a total of fifteen flat pieces for the basal structure, which were to be welded to a large central cylinder.

My job was to ‘finish’ the plasma-cut pieces: I refined the curves with an angle-grinder, drilled the holes that created the delicate openwork patterns, sanded everything to smooth the edges, then repeatedly heated and quenched each piece at the forge to darken and strengthen them. When Dave first placed the angle-grinder in my hands I was terrified—the sheer speed and power and noise of the monstrous thing connecting with the steel with a shower of sparks had a brutality and violence about it that I had never before experienced. But once I got used to the heightened energy of the process it became utterly enthralling. The grinder began to feel as fluent and expressive as a brush, and the steel felt responsive and alive. Like all metalwork processes, it demands a total, immersive concentration—you can get lost in it for hours!

Ricky Lee Brawn worked with her to make the brass parts:

Below you can see the brass piece he’s making, called a finial, among the steel legs of the partially finished harmonograph:

There are three legs, each with three feet.

The groups of three look right, because I conceived the entire structure on the basis of the three pendulums working at angles of 60 degrees in relation to one another (forming an equilateral triangle)—so the magic number is three and its multiples.

With three pendulums you can generate more complicated generalizations of Lissajous curves. In the language of music, three frequencies gives you a triplet!

Things become still more complex if we leave the linear regime, where motions are described by sines and cosines. I don’t understand Anita Chowdry’s harmonograph well enough to know if nonlinearity plays a crucial role. But it gives patterns like these:

Here is the completed harmonograph, called the ‘Iron Genie’, in action in the crypt of the St. Pancras Church:

And now, I’m happy to say, it’s on display at the Museum of the History of Science, where we met in Oxford. If you’re in the area, give it a look! She’s giving free public talks about it at 3 pm on

• Saturday July 19th
• Saturday August 16th
• Saturday September 20th

in 2014. And if you can’t visit Oxford, you can still visit her blog!

### The mathematics

I think the mathematics of harmonographs deserves more thought. The basic math required for this was developed by the Irish mathematician William Rowan Hamilton around 1834. Hamilton was just the sort of character who would have enjoyed the harmonograph. But other crucial ideas were contributed by Jacobi, Poincaré and many others.

In a simple ‘lateral’ device, the position and velocity of the machine takes 4 numbers to describe: the two pendulum’s angles and angular velocities. In the language of classical mechanics, the space of states of the harmonograph is a 4-dimensional symplectic manifold, say $X.$ Ignoring friction, its motion is described by Hamilton’s equations. These equations can give behavior ranging from completely integrable (as orderly as possible) to chaotic.

For small displacements our lateral harmonograph about the state of rest, I believe its behavior will be completely integrable. If so, for any initial conditions, its motion will trace out a spiral on some 2-dimensional torus $T$ sitting inside $X.$ The position of pen on paper provides a map

$f : X \to \mathbb{R}^2$

and so the spiral is mapped to some curve on the paper!

We can ask what sort of curves can arise. Lissajous curves are the simplest, but I don’t know what to say in general. We might be able to understand their qualitative features without actually solving Hamilton’s equations. For example, there are two points where the curves seem to ‘focus’ here:

That’s the kind of thing mathematical physicists can try to understand, a bit like caustics in optics.

If we have a ‘twin elliptic pendulum harmonograph’, the state space $X$ becomes 8-dimensional, and $T$ becomes 4-dimensional if the system is completely integrable. I don’t know the dimension of the state space for Anita Chowdry’s harmonograph, because I don’t know if her 3 pendulums can swing in just one direction each, or two!

But the big question is whether a given harmonograph is completely integrable… in which case the story I’m telling goes through… or whether it’s chaotic, in which case we should expect it to make very irregular pictures. A double pendulum—that is, a pendulum hanging on another pendulum—will be chaotic if it starts far enough from its point of rest.

Here is a chaotic ‘double compound pendulum’, meaning that it’s made of two rods:

### Acknowledgements

Almost all the pictures here were taken by Anita Chowdry, and I thank her for letting me use them. The photo of her harmonograph in the Museum of the History of Science was taken by Keiko Ikeuchi, and the copyright for this belongs to the Museum of the History of Science, Oxford. The video was made by Josh Jones. The image of a Lissajous curve was made by Alessio Damato and put on Wikicommons with a Creative Commons Attribution-Share Alike license. The double compound pendulum was made by Catslash and put on Wikicommons in the public domain.

### 35 Responses to The Harmonograph

1. anitachowdry says:

Hi John, I feel unbelievably honoured by this coverage of my work, with additional in-depth comments about the history and mechanics of the harmonograph – “thank you” hardly seems adequate!

It was very inspiring to make that first visit to the Oxford Museum of the History of Science and to meet you there, just as I was submitting my proposal to display the Iron Genie at the museum. The two lateral pendulums work together to drive the pen, so can move it in any direction, and the drawing platform pendulum is mounted on a gimbal, so it can swing in any direction too – the equations are a bit beyond me though!

On the Blackburn pendulum, there is an artist called Paul Wainwright who has constructed something very similar, which he uses to create beautiful photographic images, – you can see the images and a video about how he sets it all up at

Deeply gratified by your interest, and thank you so much!

• John Baez says:

I’m honored that you let me use your images for this article. Alas, my article is intangible and virtual, though it’s about something very solid and real.

It sounds as if the state space of your harmonograph (which I’ve never seen in person) is 8-dimensional. I believe it takes 4 numbers to specify the position of the pendulums:

The two lateral pendulums work together to drive the pen, so can move it in any direction, and the drawing platform pendulum is mounted on a gimbal, so it can swing in any direction too.

So, 1 number to describe the angle of each lateral pendulum, and 2 numbers to describe the position of the drawing platform pendulum, for a total of 1 + 1 + 2 = 4 numbers. (An engineer might say ‘4 degrees of freedom’.)

If so, the ‘state space’ of the harmonograph is 8-dimensional, since for each number to describe its position we need another to describe its velocity.

So, the curves it draws are really curves in an 8-dimensional space, projected down to the 2-dimensional sheet of paper!

2. John Baez says:

I just added a comment:

The harmonograph is now on display at the Oxford Museum of the History of Science—and Anita is giving free public talks about it at 3 pm today on Saturday July 19th, and also Saturday August 16th and Saturday September 20th!

If you’re near Oxford, check it out!

3. One great thing about harmonographs is that they can make stereograms– 3D instead of 2D projections of higher dimensional curves. This fact was discovered over a hundred years ago.

• John Baez says:

Can you explain a bit more?

• I read about it in an article by Charles Benham when I was doing research for my book. If you set the harmonograph to create “two figures identical in size and period, but slightly differing in phase” then you can use a stereogram viewer (or just cross your eyes) to see a 3D curve.
I just now made one using an online harmonograph with sliders.

• Benham also experimented with setting a bright spot vibrating while taking two simultaneous long exposures from different perspectives, to create similar effects photographically.

• anitachowdry says:

Hi Douglas – I have bought and read a copy of your book “Machinamenta” which I found very interesting.

I can’t quite get a true stereoscopic effect from your example, but have seen several of the old photographic examples—usually viewed through a special viewer. Trouble with an analogue machine is, you can never set up the parameters to reproduce themselves exactly—so very hard to predict the outcome of each drawing!

4. HenryB says:

I think Miss (Mrs) Chowdry needs another grant. Her work is only half finished. What about a harmonograph built out of, not simple, but rather cycloidal pendulums? I’m wondering about the apearance of graphs sketched out by a device able to deliver motion which solves the problem of the tautochrone.

5. John Baez, you are wrong about this “the frequencies aren’t perfectly tuned, so we get more complicated Lissajous curves.”
Frequencies are perfectly tuned. You get those figures if the amplitude of one pendulum decreases faster than amplitude of the other pendulum.

See Mathematica code:

delta = 0.02;
lam = Exp[-delta t];
w1 = 2 Pi;
w2 = 3/2 2 Pi;
phi1 = 0;
phi2 = 0;
A1 = 1 lam;
A2 = 1 lam^2;
X[t_] := A1 Sin[w1 t + phi1];
Y[t_] := A2 Sin[w2 t + phi2];

ParametricPlot[{X[t], Y[t] }, {t, 0, 40}]

• John Baez says:

Elena wrote:

Frequencies are perfectly tuned. You get those figures if the amplitude of one pendulum decreases faster than amplitude of the other pendulum.

Nothing is ever perfect, but I agree that your explanation is a good one. No matter how perfectly you tune the frequencies, it’s likely that friction will affect one pendulum more than another. And even if it affects them both the same way, the Lissajous-like curve will ‘shrink’ as time passes, making a more complicated pattern.

• I can reproduce the curves from the old book exactly.
http://elena-mur.livejournal.com/65611.html
Can you with your explanation?

One can see right from the pictures that the frequencies are well tuned and friction affect pendulums differently. It is all in the link above.

• John Baez says:

Elena wrote:

I can reproduce the curves from the old book exactly.

Nice! I already said your explanation was good; after viewing your pictures I’m willing to say it’s right.

• anitachowdry says:

I am impressed by the images! I shall be scanning and posting the entire book on my website soon – would be interested to see more interpretations…

• John Baez says:

When you put the book on your website, it would be great if you could include a link here, so harmongraph fans can have another way to find it. I’m looking forward to reading it!

I also find that mathematical instruments deserve more attention, so it nice to see that there are actions to construct them again!
(Here is a DIY version of what looks like the above harmonograph).

And very interesting in this context is the so-called integraph by
Bruno (Abdank)-Abakanowicz (who invented also the spirograph). The integraph (a more detailed description in german though is here) seems to be very rare, in particular in the book L’Europe Mathématique: Histoires, Mythes, Identités it was written:

These instruments – particularly the integraph – may be used to mechanically calculate an area delimited by curves such as intervene in the determination of the work performed by a steam engine (integrating around work cycles). Unfortunately, I don’t know where these instruments are currently located.

It seems one integraph was sold in 2006.

Finally mathematical drawings seem also to inspire art historians, like at the Warburg library.

• John Baez says:

Here are some integraphs from the Wikipedia article:

I’m not sure what’s the difference between an integraph and a planimeter. I guess with a planimeter you trace a closed curve that’s the boundary of a surface whose area you want to know, while with an integraph you trace along the graph of a function you want to integrate.

7. anitachowdry says:

There are some interesting mechanisms stored away at the Museum of the History of Science Oxford, I am told! Maybe they have something like this hiding somewhere… I think we might get a chance to see some, at some point!

• Walter Blackstock says:

Pendulum autographs.
Hubert Airy
Nature, 1871, 310-313 (part I) and 370-372 (part II)

“It was a happy chance that directed my fingers, in an idle mood, one day in March of last year, to the top of a stiff twig that sprang from the stool of an old acacia, and rose to a height of about three feet, where it had been lopped by the gardener’s knife. Pulling the twig aside, and letting it fly back by its own elasticity, I noticed the path which its top traced in the air. … On the present occasion I could see that the twig began at once to deviate from the plane of its first vibration, and to describe an elliptic path, the ellipse growing wider and shorter till it was nearly circular, then still wider and still shorter, till its width exceeded its length, and it was again elliptic, but the long axis now occupied nearly the position of what was the short axis before”.

Moving from the garden to the workshop (his bedroom – it would have a high ceiling) he improvised, risking half-a-hundredweight of lead going through his bedroom floor and struggling to make a reliable pen. Finally: “It chanced, however, that the adjustment for the proportion 2 : 3 was beautifully accurate I shall never forget the feeling of delight which I experienced while watching the marvellous fidelity with which the pen point traced the curve appropriate for that proportion”.

I think Anita would recognise that feeling. Airy senior would have appreciated the Mathematica code too.

https://archive.org/stream/nature41871lock#page/310/mode/2up

Marvellous expansive writing that would not stand a chance today, especially in Nature!

Warning: reading old issues of Nature can be addictive.

Hubert Airy (1838-1903), a physician, was a son of Sir George Airy (1801-1892), mathematician and Astronomer Royal (1835-1881).

I think we might get a chance to see some, at some point!

Who is “we” ? – museum wards, artists, students, the british upper class?

In the Math building in Göttingen there were (at least some years ago) instruments visibly on display. The exhibition was in glass boxes and was open whenever the math building was open. The collection holds also a reconstruction of the Integraph by Abdank-Abakanowicz Coradi, reconstructed by Mühlendyck.

8. Postscript: Check out John Baez’s article about this, with mathematical analysis, on His Azimuth blog post ‘The Harmonograph’ […]

9. There was a toy version of this. I got one (in Chicago) when I was maybe nine years old (late 1960s). It was marketed as a step up from the “Spirograph”, which was then fairly new on the market. It was flimsy — nothing like the huge, stable machine that you show! There was a plastic frame that clamped to the edge of a table. The pendula were red plastic rods weighted with bags of water — one of which very soon burst. I had forgotten about it from that day to this.

10. anitachowdry says:

I have now created a page of harmonograph resources on my website, including some of the links suggested by commenters on this page. I have also uploaded a PDF of my 1893 book “Harmonograph” by H. Irwine Whitty (with the illustrations that were referenced by Elena Murchikova, above) which you can view or download:

11. David Winsemius says:

Friction, which I initially interpreted to mean loss of kinetic energy (dissipation) was mentioned above as an “issue”. One of the links that came up on the YouTube search mentioned that friction was the mechanism for coupling of the harmonic oscillations between the pen-pendulums and the platform pendulum. That could tie in to the El Nino-Southern Oscillation question by way of the fact that there would be a coupling of (low frequency) cold oceanic currents to the (high frequency) atmospheric currents.

12. anitachowdry says:

Hi John, for those who are interested, I have just posted a new article about the archives at the Museum of the History of Science Oxford, which includes inventors of the geometric chuck and a rare stereoscopic harmonograph drawing (1893) which you can download: http://anitachowdry.wordpress.com/2014/10/06/ingenious-machines-for-drawing-curves-the-archives

13. anitachowdry says:

Hi John,

Apologies to anyone who has recently tried to access this post and found it password protected—I have temporarlly withdrawn it from the public domain while we consider some possible copyright issues in respect of images from the museum archives. Hopefully it can be resolved fairly soon.

14. anitachowdry says:

I have just been given the go-ahead to reinstate the post by the museum archivist – he is happy that proper acknowledgement has been given to the museum, and that the material is presented within the context of scholarship. Apologies to anyone who was frustrated by the inaccessibility of the article – it is now freely available to read & comment!

• John Baez says:

Great! I’d already read the article, but I hope more readers here give it a try now.

15. lrconsiderer says:

Gorgeous, gorgeous, gorgeous. Thank you so much for such a detailed post :D

16. […] The harmonograph is also known as the first art machine–because it is created with something non-human. It uses basic physics, a pencil, and a piece of paper, to produce a large variety of beautiful designs. […]

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