Epicycles

a simple black line diagram on a white background. it's a loop that folds in on itself in 3 places, and then that folding in also fold back on itself

I have just released a new tool for generating epicycles.

If you’re wondering what an epicycle is, here’s the short version: it’s a pattern like a Spirograph might make. You know, those toys you might’ve played with as a kid. Where you put a pen tip in a plastic cog and then spin it around inside a larger plastic cog and then it makes a nice shape on the paper.

The longer version goes back to the motion of the planets. Before it was widely agreed that they orbited the Sun, people would still observe them and try to predict their movements. People have always wondered about the sky, and while most of it stays in place relative to itself, there were some bright spots that moved within a complicated pattern. Usually the planets would move forwards across the sky, but sometimes they would go backwards for a bit (the term for this is retrograde motion, which you might recognise from horoscopes). So, while inventing the concept of geometry, the ancient Greeks found that they were able to describe this motion by tracing the way a point would move if it was on a circle that was itself moving around a larger circle. Mostly it goes forwards, but sometimes (when the smaller circle is turning against the larger one) it goes backwards. And that’s what an epicycle is! Wheels within wheels.

There’s a lot of history I’m skipping here, but let me gesture to it with this diagram from Johannes Kepler’s 1609 treatise Astronomia Nova, which plots out the course of the planet Mars (as recorded by Tycho Brahe). The kind of shape it describes will become familiar to you if you spend a few minutes playing around with this tool.

it's a scan of a very old book - a diagram with a circle, and beautiful loops going in to the center of it. there are dates annotated on it at two points, and the outer circle is surrounded by the signs for the constellations

So! This tool is a tool which will let you visualise the path taken by a point controlled by the motion of many circles combined together.

Each circle within the tool is defined by three parameters:

And you can also toggle on a mode where the phase changes over time.

But that’s it! But within that, I find it pretty remarkable the range of forms it can create. I love making tools that provide a set of simple parameters, but which can form a surprising variety of outputs. Not least, because I like to explore them myself - and even now I’m releasing this tool, I don’t think I’m finished finding new things within it.

I originally started investigating epicycles in order to generate files to plot with my pen plotter - and this tool is still useful for this, in that pressing the button in the corner will download a SVG of the currently displayed epicycle. You are free to use these SVGs to do whatever you like with - plot them, print them, use them as the basis for a art or graphic design. Credit for using a neat tool is appreciated, but ultimately this is just maths & as such belongs to everyone.

All of which sounds very serious - but really, this is a tool for messing with. Push some buttons, move some sliders, get a feel for it, and see what interesting loops you can find. Go play!


02 August 2022