Solitons, platonic symmetry and fullerenes

Skyrmions are topological solitons in three space dimensions which are candidates for an effective description of nuclei. It is of interest to determine the structure and symmetry of the classical Skyrmion solution with minimal energy for a given soliton number. Numerical solutions of the relevant nonlinear PDE yield interesting results, with the Skyrmions often having unexpected Platonic symmetry. For large soliton numbers the solutions have a structure similar to Fullerenes in carbon chemistry. These results are reviewed, together with an analytical perspective which involves an approximation based on rational maps between Riemann spheres.