In response to the Sphere project, regular visitor Eric sent me links to his brother's flickr feed and website where he shows how he projects panoramic pictures onto for various 3D solids. Fantastic stuff!
I've followed the link from his site and downloaded the demo of Flexify which I'll be purchasing shortly.
With Flexify I can take their demo picture (above) and convert it so that it covers a variety of different 3D shapes from cubes through dodecahedrons (below) to the truncated isocahedron from the sphere project.
Cut out the net, glue it together. (You could right click the above picture, save it and print it out to try this at home) The result is this fascinating 3D projection.
So, next step, create some of my own panoramic pictures to stitch together and project. Thanks Eric!