Long, long ago, when the Internet was young and so was I, Compuserve ran a successful collection of forums on many and varied topics. Their Science & Maths Forum in particular brought me delight and illumination; reading lucid explanations of difficult topics, and attempting to produce them myself, had a great formative influence on me as a potential scientist and educator. There’s nothing quite like explaining something to get it straight in your own head.

One of the internet-friends I made there was an American named Jeff Werbock, who came bearing a vision of the universe as a unified whole composed of space multiply curved in on itself, and forever converging and diverging: a sort of universal hypertoroid. Though I’m not convinced that it entirely captures the way the cosmos works, I very much like its focus on curvature and movement as fundamental features of the structure of spacetime, and I think it has great aesthetic appeal.

So when it struck me one day that I knew exactly how to create a visualisation of it using POCO, the animation-based C variant that I was programming in at that time, I was pleased to sit down and make it. This was one of the first mathematical animations I ever created, and the interactive version you see here is on extremely similar lines.

We start by drawing a toroid, which is a circle swept out in a circle, like a geometrically perfect bagel. The circles that make up the toroid are rotating, so the toroid is forever converging and diverging. Dragging the mouse left or right changes the degree to which this toroid is twisted around on itself. This gives us a rough approximation – or at least a cross-section – of a hypertoroid, which is a toroid swept out in circle.

Drag the mouse up and down in the frame to change the size of thing, and hold down the shift key while you do it to change the minor radius independently from the major radius.

I am currently working on a mobile version of this using Processing for Android, and a narrative version to accompany Jeff’s in-depth explanations of how he sees this all working…

Most of them are variations on applets I first created some time ago. The impetus to collect them all together in one place came when I set about creating an interactive installation, Kenneth and the Waves, using a more elegant set of inputs than that allowed by a mouse and keyboard.

The first shows the most basic interaction of two waves – the black wave at the top is the superposition of the waves, which is to say it’s what you get when you add them together. When the two waves are in phase, the peaks of one are added to the peaks of the other; when they are in anti-phase (half a cycle out of synch with each other) they cancel out. Among other things, this is the basis of interference, and standing waves – which are what you get when two identical waves travel in opposite directions in the same space. Standing waves don’t travel at all, even though they are made up of waves travelling in opposite directions. All musical instruments make use of standing waves, and they are important for various other reasons.

Since all the animations are all based on pairs of simple waves, they can all be controlled by changing the frequency, amplitude and speed of those waves. To do this, drag the mouse cursor in the animation box to change frequency (left button), amplitude (right button or shift-click) and speed (middle button or ctrl-click). The vertical axis controls one wave, horizontal axis controls the other. Press a key to switch between animations.

If you don’t have a working Java plugin for your browser, or you just want a bigger version, you could try downloading one of these applications for your operating system:

• Linux (32-bit)
• Linux (64-bit)
• Mac OS X
• Windows (32-bit)
• Windows (64-bit)

Since I planned to travel all over Spain and Portugal carrying this thing, I figured I’d better make something more portable than the beautiful fire-engine-red box that Tom Hardiment made me for Kenneth Mark I, which was originally a solid hardwood drawer. I took the opportunity to give it a completely different aesthetic.

I was always torn between two looks for controls for my animations – either they should be hyper-futuristic, like a starship control panel, or stained wood and brass like the machines and instruments of the Victorian era. I have always had a thing for antique scientific equipment, so I can understand why steampunk has become such a big thing. I decided to go with wood and brass.

I stripped out the plywood with the sliders and buttons attached, cut it down to size, sanded it, stained it and varnished it, and I picked up some brass knobs  and edging. Then I travelled with it to the Basque Country, where I was teaching English and looking after kids on a summer camp for a couple of weeks.  One of my colleagues there, Gareth Fox, helped me put it all together with some support struts; I replaced the plastic slider knobs with the sort you might otherwise use to open small doors. Then I went travelling again, right across to the southwest of Spain and then back up through Portugal.

Unfortunately somewhere on that largely unplanned journey (it’s a long story) I  misplaced the Arduino board that provides the interface between the sliders and a computer’s USB port. It’s not always easy to find an Arduino in a small city in a foreign country, and with no local knowledge at my command I googled ‘Arduino Coimbra’ and reached out for advice to the one person I found listed on the Arduino site, Micael Couceiro of RoboCorp and ISEC.  He suggested that if I didn’t have time to order online and wait for a delivery, I should try a shop on the outskirts of town, so I went there at lunch the next afternoon and sat around for about two hours waiting for them to open, only to find they were all out.

Eventually he very kindly agreed to lend me a board from his department, so I went over there the next day and in the end he didn’t just lend me a board but he and his colleagues offered an amazing amount of support and practical help mounting it properly and putting finishing touches to the box.

I finally got to exhibit Kenneth Mark II properly on the very last day of the conference – a little late, but not too late for a whole lot of people to play with it and give very positive feedback!

I figured that what’s really wanted is a bank of sliders and buttons to play with, each controlling a parameter. That way people can walk up to the controls as if they were approaching the bridge of a starship, and just twiddle them however they feel.

Now, with the help of the Dorkbot Alba team, I finally have the controls I’ve long dreamed of, in the shape of a box with six sliders, five buttons and two glowing switches. It is a thing of beauty.

The obvious next step once you have a good interface for a set of generative animations is to start putting them in public places for people to experiment with, I think.

The first such place was the Forest Hall in Edinburgh, at a night called Dubversion, along with a collection of electrical strangeness from Dorkbot including the ‘Waldflöte‘ MIDI-rigged pipe organ, a Jacob’s Ladder and the amazing Quadracopter – as well as some excellent music.

I’m seeing this as an installation at a science museum, in the classroom or an art gallery. There’s lots more still to do with it.

I decided to make sure the sliders control much the same things in different animations – though they each look very different, they are all built out of trigonometric functions, so we can think of them as the product of waves. The most basic properties of a wave are its amplitude (loudness, if we’re talking about sound); its frequency (the pitch of a sound) and the speed at which it travels. Though they do very different things with them, each of the animations I’ve rigged up with this can be seen in terms of the two sets of interacting waves, so we have a pair of sliders controlling their amplitudes, a pair for frequency and a pair for wave speed.

As I’ve written elsewhere, waves are absolutely fundamental to an enormous range of physical processes, as well as being great fun to animate. The universe is pretty much made out of waves, so wave speed, frequency and amplitude are three of the most basic and far-reaching concepts in physics.

Applets adapted to work with the control panel:

• Shimmia (with added webcam input)
• Zoobie
• Rosaly
• Yinyo
• (also Trochor, finally re-done in Processing – more on that later – and Dragoria, but that doesn’t work with compatible controls…)

Credits:

• Tom Hardiment built the slider-box.
• Martin Ling and Al Bennett helped out with the wiring and setting up the Arduino unit.