Visual Insight

1 March, 2015

I have another blog, called Visual Insight. Over here, our focus is on applying science to help save the planet. Over there, I try to make the beauty of pure mathematics visible to the naked eye.

I’m always looking for great images, so if you know about one, please tell me about it! If not, you may still enjoy taking a look.

Here are three of my favorite images from that blog, and a bit about the people who created them.

I suspect that these images, and many more on Visual Insight, are all just different glimpses of the same big structure. I have a rough idea what that structure is. Sometimes I dream of a computer program that would let you tour the whole thing. Unfortunately, a lot of it lives in more than 3 dimensions.

Less ambitiously, I sometimes dream of teaming up with lots of mathematicians and creating a gorgeous coffee-table book about this stuff.


Schmidt arrangement of the Eisenstein integers


Schmidt Arrangement of the Eisenstein Integers - Katherine Stange

This picture drawn by Katherine Stange shows what happens when we apply fractional linear transformations

z \mapsto \frac{a z + b}{c z + d}

to the real line sitting in the complex plane, where a,b,c,d are Eisenstein integers: that is, complex numbers of the form

m + n \sqrt{-3}

where m,n are integers. The result is a complicated set of circles and lines called the ‘Schmidt arrangement’ of the Eisenstein integers. For more details go here.

Katherine Stange did her Ph.D. with Joseph H. Silverman, an expert on elliptic curves at Brown University. Now she is an assistant professor at the University of Colorado, Boulder. She works on arithmetic geometry, elliptic curves, algebraic and integer sequences, cryptography, arithmetic dynamics, Apollonian circle packings, and game theory.


{7,3,3} honeycomb

This is the {7,3,3} honeycomb as drawn by Danny Calegari. The {7,3,3} honeycomb is built of regular heptagons in 3-dimensional hyperbolic space. It’s made of infinite sheets of regular heptagons in which 3 heptagons meet at vertex. 3 such sheets meet at each edge of each heptagon, explaining the second ‘3’ in the symbol {7,3,3}.

The 3-dimensional regions bounded by these sheets are unbounded: they go off to infinity. They show up as holes here. In this image, hyperbolic space has been compressed down to an open ball using the so-called Poincaré ball model. For more details, go here.

Danny Calegari did his Ph.D. work with Andrew Casson and William Thurston on foliations of three-dimensional manifolds. Now he’s a professor at the University of Chicago, and he works on these and related topics, especially geometric group theory.


{7,3,3} honeycomb meets the plane at infinity

This picture, by Roice Nelson, is another view of the {7,3,3} honeycomb. It shows the ‘boundary’ of this honeycomb—that is, the set of points on the surface of the Poincaré ball that are limits of points in the {7,3,3} honeycomb.

Roice Nelson used stereographic projection to draw part of the surface of the Poincaré ball as a plane. The circles here are holes, not contained in the boundary of the {7,3,3} honeycomb. There are infinitely many holes, and the actual boundary, the region left over, is a fractal with area zero. The white region on the outside of the picture is yet another hole. For more details, and a different version of this picture, go here.

Roice Nelson is a software developer for a flight data analysis company. There’s a good chance the data recorded on the airplane from your last flight moved through one of his systems! He enjoys motorcycling and recreational mathematics, he has a blog with lots of articles about geometry, and he makes plastic models of interesting geometrical objects using a 3d printer.

The Harmonograph

18 July, 2014

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:


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.