A multi-part theorem is presented concerning the morphogenesis of high-symmetry structures made of three-dimensional morphological units (MU's) free to move on the surface of a sphere. All parts of each MU interact non-specifically with the remainder of the structure, via an isotropic function of distance. Summing all interactions gives a net figure of merit, ℐ, that depends upon MU positions and orientations. The structure evolves via gradient dynamics, each MU moving down the local gradient of ℐ. The analysis is reresented with generality in Fourier space, which eases the expression of symmetry. Structures near symmetry, but far from a local minimum of ℐ, are analyzed. For each, a symmetrical configuration can be found, for which ℐ is an extremum with respect to symmetry-breaking perturbations. Under gradient dynamics, a quadratic measure of such deviations from symmetry decreases monotonically, anywhere in the large basin of attraction of a local minimum. Thus: high symmetry is an attractor. Application is made to icosahedral virus capsids. The Symmetrization Theorem shows that a stable capsid, maintained by non-specific interactions among its capsomeres, could arise generically in a "bottom-up" process. For animated evolutions that self-assemble into high symmetry, visit
Pattern formation in icosahedral virus capsids: the papova viruses and Nudaurelia capensis beta virus
