Set your calculator to “maths”…

Yesterday two basic criteria for our New Year’s LED project were established:

1. Our “ball” will actually be an icosahedron — a 20-sided shape — for ease of assembly. Each face will have six LEDs, for 120 LEDs total.
2. To create meaningful patterns on the ball, rather than just random blinking, it’s necessary to know the coordinates of every LED. Therefore, a software model will need to approximate the physical thing.

The first step to finding the LED coordinates is to establish the positions of each of the 12 vertices of the icosahedron, even though there are no LEDs there. Once those points are known, all the LEDs can be located through interpolation.

There are a number of ways of constructing an icosahedron…the Greeks were working up the math in 400 B.C., the Scots might have hammered it out in the late Neolithic Age…though some protozoa and viruses had us all beat by half a billion years or so.

Our “ball” will hang from one vertex at the top, so we’ll follow a construction method to suit. When viewed down the vertical axis, there’s a clear 5-fold symmetry:

Starting with just the radius of those “spokes” from the center (we’ll use a value of 1.0 — the units are arbitrary), a couple formulas are all that’s needed to find the 3D position of one vertex, and from that to replicate all the remaining vertices by revolving around the center axis. Wikipedia can help with the maths!

From those vertices we can then identify edges and faces. Using two edge vectors from each face, the triangular 1-2-3 layout of the LEDs can be interpolated across this. Wrapped up in just a short Processing sketch, we now have a tumbling 3D icosahedron with LED dots in the desired places! Later, we’ll issue colors based on each LED’s location.

A test sketch for Processing is available from the project’s Github repository. This does not yet communicate with an Arduino or the LEDs; it is strictly for validating the geometry. It’s a good visual match for the model on the table!