Shimmia Zoobie Yinyo SnakeCharmer Resonata Squared Rosaly Dragoria Curlicor Dragoric Curlicue Fractal Quadrina Toroidia Net of Indra Trochor


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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…

Kenneth and the Waves

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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.

Trochor – Animated Virtual Harmonograph

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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.

An Introduction to Programming, Using Processing. Part 1: Why Program?

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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.


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Algorithmically distorting images is fun! Shimmia re-uses most of the maths and the code from Zoobie – both programs run through a grid of points, and for each one they pick a second point, displaced from the first by the interactions of waves.

Zoobie then draws a semi-transparent triangle at the second point, whereas Shimmia draws a pixel at the first point which gets its colour from second point in an image. The relation between them is quite a lot like the one between caustics and refracted images, though the mathematical analogy is not exact.

Click and drag inside the applet with either mouse button (or, if you only have one, with and with the closest thing you have to a Ctrl button) to change the frequency and amplitude of the waves. Drag with shift held down to change their speed.

For now the image is just this one I took of some leaves, but I figure I’ll see if I can pull pictures from Flickr later.

Snake Charmer

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Snake Charmer is an applet closely based on Paul Friedlander’s light sculptures, which use hanging ropes spun at varying speeds, combined with ‘Chromastrobic’ lights’ (changing colour very quickly), to produce stunning visual effects. I recommend seeing the full-sized, physical ones if you ever get the chance, but I think there’s a lot of fun to be had with this interactive simulation, too.

Resonata Squared

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Resonata Squared is based on a stack of chains, each transmitting waves at a different speed, but all driven by the same driving frequency.

The original Resonata is considerably more sophisticated in various ways, but really almost completely different.

Big images

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Having finally looked up how to produce big images in processing, I’ve been having great fun digging up code I originally wrote more than a decade ago in POCO (the Autodesk Animator Pro variant of C), translating it into Processing and whipping up stills which are potentially big enough to print at about two feet across (I’m settling for 2400×2400 for now – nothing seriously vast). Here’s a sampling that you can view at 1024×1024, or here’s a full-size version of the first


Marillos 1Vorticoid 03ix08d

Vorticoid 7
Vorticoid 4
Vortical 1
Vortical 2
Vortical 3

I might make t-shirts of some of these (etc.) available in my Cafepress Trigonometry shop (already selling some related designs), although I’ve never really made enough sales there to justify the effort.


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Curlicor is a very simple toy. You basically control the steering wheel of the triangle. If you keep turning it you produce curlicue fractals. Click on the applet to toggle between mouse and keyboard control. Unlike most of my interactive animations, this one really requires interaction in order to do anything interesting.