Blowing Carbon Bubbles : Expanding 2D Fullerene Layouts to 3D

The concentric face layout code is working well enough now to handle the larger fullerenes - such as that old favourite, C60. Since coloring the vertices by equivalence class is not always terribly informative, here is a view of the ring equivalence classes :

Where C60 is on the left, and a more colourful C70-D5h is on the right. One difficulty, however, is to understand the symmetries of these structures when they are distorted like this. The further away from the center of the layout, the more stretched the rings become.

So, an obvious next step was to 'blow up' these 2D layouts into 3D. It turns out that is possible, with a combination of inverse stereographic projection and Jmol's minimize command. The first step is necessary since minimizing the 2D coordinates (with a z-coord of zero) just shrinks the diagram down in the plane. Here are before and after shots of these steps:

Clearly the inverse-projection does not give very good 3D positions for the atoms, but they are on the surface of a sphere, and the bonds don't cross. The minimization gives a much better looking version, although there are still dents and a lot of asymmetry.

Next step is to write out the equivalence class colours (vertex/face?)  into a Jmol script, so that they can be visualised in 3D. Oh, and one last thing - it was necessary to scale the flat diagram down so that the radius was closer to a unit circle. If it was contained inside the circle, then the expansion was only a hemisphere. Also, the points were expanded from the centre outwards by a small factor, to improve the bond distances.