# Explanation of Stellation

For further information see 'The Fifty-Nine Icosahedra' by H. S. M. Coxeter et al. Tarquin Publishing. ISBN 1-899618-32-5

According to the dictionary, to stellate simply means to make star shaped (i.e. stellar).

However there is also a more precise definition as applied to polyhedra:-

Stellation is the process of extending the faces of a polyhedron until it intersects with another extended face of the same polyhedron.

We'll concentrate on the icosahedron, but the following steps could be applied any polyhedron.

• First get your polyhedron. You can make an icosahedron using the instructions here.
• Now place the icosahedron on a large piece of paper. Make sure that it won't move.
• Now take a piece of card and lay flat against one face of the icosahedron such that one edge lies flat on the sheet of paper. Draw a line along this edge. You have now drawn the intersection between the bottom face extended and the other face extended. In other words you have drawn the edge at which these two faces can meet.
• Now repeat for the other 18 faces (or whatever if you are using a different polyhedron). Note that in the icosahedron (and many other polyhedra) one face will be parallel to the original and will therefore never intersect.

The final shape (ignoring lines that wander off into infinity) looks like this:

This is a map of all the intersections of that particular face. This means that the regions between the lines are potentially the bits of this face that you might see in the stellation. And of course with regular polyhedra (like the icosahedron) all the extended faces will have the same regions as they are all the same shape.

The long outside points only appear in one stellation (number 8) and are usually not shown. The pattern that is left is the one used for the region map and also for the background of this site!

So now imagine picking some of the Regions and colouring them in. Take this pattern and lay it on every face of the icosahedron (where the face matches the central triangle). You have a potential stellation of the icosahedron.

But the edges may not join, or the areas will try to cut through each other, or it won't have the same sorts of symmetry as the icosahedron or it won't form closed groups of faces.

By applying these strictures (defined in more detail by Coxeter et al.), we discover that there are 59 possible groups, or region maps, and therefore 59 stellations.

The numbering scheme used on this site follows Coxeter's numbering. There are other schemes, so don't be surprised if the number shown here is not the one that you are used to.

Each region map will be shown with the appropriate Stellation. If you want to find out how to decode the region maps, follow this link:-