The picture displayed is the convex hull of the points of a potential minimizing configuration. Our algorithm for generating the convex hull determines the faces by testing each threesome to see if all other points lie on the same side of the plane the three points generate. The figure is generated by then showing all of the faces (or what is not obscured by Mathematica's hidden surface algorithm).
We display one view of the configuration after having run an animated flight around the configuration and chosen a view that might reveal most of the structure. Naturally, some of you will want to see more of the configuration. We hope to provide a copy of the Mathematica code which generates the views so that you can try your own views. If we missed something nice please let us know.