The Chrono-Synclastic Curlicue Clock shows three Curlicue fractals, corresponding to the current hour, minute and second of the day. With practice you should be able to tell the time with it – every time of day is associated with a specific shape of Curlicue.

At particularly notable times of day, the Curlicue dramatically simplifies, either by folding right in on itself, or reaching way out towards infinity. This is because the complexity of the seed, expressed as a continued fraction, relates to the complexity of the corresponding Curlicue. You can see this happen with the red-orange ‘second hand’ curlicue at various points throughout each minute. The longer-cycle fractals take longer, but they move about as fast because they show more of the fractal form. In the build-up to six o’clock, you can see the blue ‘hour hand’ curlicue spinning ever-inward, folding further and further in on itself, until at the exact moment it turns six – when the seed of the fractal is precisely one half – it collapses to just two points.

The most complex forms come when the seeds are irrational.

Here is another version. It has just one curlicue on a 24-hour-cycle, together with standard clock hands.

‘Chrono (kroh-no) means time. Synclastic (sin-class-tick) means curved towards the same side in all directions, like the skin of an orange.’

- A Child’s Cyclopedia of Wonders and Things to Do

This is an animation based on toroids, and what happens when circles of circles go in circles of circles: an image of endless four-dimensional convergence.

This is how it came about…

This is a set of visualisations which are all, in one way or another, based on pairs of interacting sine waves.

It turns out you can do a lot with sine waves…

This is an interactive diagram of a guitar fretboard. It shows you all the different ways that chords can be made, in different tunings.

Here is what that means, and why it exists…

This summer I went travelling around the Iberian Peninsula, partly because I had some of my work accepted for the exhibition accompanying the 2011 Bridges Conference on connections between art and mathematics, in Portugal – specifically, a large print of my generative art still ‘Vortical‘ and my interactive installation based on waves, Kenneth (see the visualisations it works with here).

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.

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Everything we see and hear is made of waves, and the interactions of different frequencies – interference, resonance and harmony – account for many of the most interesting things there are. They are also a lot of fun to visualise, so I have put together a collection of animations – applets – which are all different visualisations of the interactions of waves, or their close cousins the circles. These are designed for display in public places, with a control box for people to experiment with and supporting literature.

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Interactive animations need intuitive controls to make them easy to play with. Since they always have a bunch of parameters to control, dragging with the mouse always seems a bit clumsy.

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. (more…)

The pattern traced out by Trochor is what you’d get if you took a pencil moving in ellipses, and used it to draw on a sheet of paper that’s also moving in ellipses. It’s a bit like a spirograph, but not constrained in quite the same ways. It’s more like a harmonograph; more on that later.

Have fun, play around with the settings, especially ‘ratio’; that’s probably the best way of figuring out what’s going on. ‘Eccentricity’, by the way, is a measure of how flattened an ellipse is – an ellipse with 0 eccentricity is a circle, one with 1 is a line.

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Processing is a language created by Ben Fry and Casey Reas of MIT, designed to make computer programming accessible to people who might imagine it will always be beyond their grasp. Processing makes it easy to create beautiful, interactive graphics.

The principles of computer programming are surprisingly simple and powerful. They also provide an easy way in to understanding some very important concepts in mathematics and science, by making them into things you can play around with. Messing about with stuff is one of the ways that people learn best.

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For those who might be interested in where all these pretty pictures and animations come from, I have recently written a piece about Processing, the open-source programming language that most of them are made in.

Processing is designed to be easy to learn and quick for knocking up simple programs which explore particular graphical and mathematical ideas. It is one of my favourite things ever, and I would encourage anybody to have a play around with it some time.