Spherical capsids are shells of protein subunits that protect the genomes of many viral strains. Although nature displays a range of spherical capsid sizes (reflected by the number of subunits in the formation), specific strains display stringent requirements for forming capsids of specific sizes, a requirement that appears crucial to infectivity. Despite its importance in pathogenicity, little is known regarding the determinants of capsid size. Still less is known about exactly which capsids can undergo maturation events such as buckling transitions-postcapsid-assembly events that are crucial to some virus strains. We show that the exclusive determinant of capsid size is hexamer shape, as defined by subunit-subunit dihedral angles. This conclusion arises from considering the dihedral angle patterns within hexamers belonging to natural canonical capsids and geometric capsid models (deltahedra). From simple geometric models and an understanding of endo angle propagation discussed here, we then suggest that buckling transitions may be available only to capsids of certain size (specifically, T < 7 capsids are precluded from such transformations) and that T > 7 capsids require the help of auxiliary mechanisms for proper capsid formation. These predictions, arising from simple geometry and modeling, are backed by a body of empirical evidence, further reinforcing the extent to which the evolution of the atomistically complex virus capsid may be principled around simple geometric design/requirements. auxiliary proteins ͉ capsid buckling ͉ deltahedra ͉ endo angle constraint A large number of human-and crop-infecting viruses are protected by spherical capsids (shells) of various sizes that are primarily made up of self-organizing protein subunits (1, 2). Caspar and Klug's (3) seminal article on quasi-equivalence explained how an infinite range of capsid sizes can be ''constructed'' by combining 60T subunits or 12 pentamers (5-valent subunit clusters) and a variable number of hexamers [10 ϫ (T Ϫ 1)] into a closed spherical shell (T ʦ {1,3,4,7, . . .} and is the triangulation number described in ref.3).From the range of possible sizes, generally, subunits from specific viral strains assemble into capsids of specific sizes; the inability to form those native sizes is believed to result in the loss of infectivity. For example, the sobemovirus and birnavirus capsids (4, 5) shown in Fig. 1A are known to be pathogenic primarily in their native T ϭ 3 and T ϭ 13 capsid forms, respectively. Despite its importance in pathogenicity, our picture of capsid size specificity is incomplete. In the present report, we are interested in the structural features (constraints), if any exist, that differentiate between capsids of different sizes (capsid design criteria). An appreciation of these concepts is pressing from a nanotechnological perspective [for the rational design of artificial scalable assemblies that build on current practices, such as in the use of protein fusion and symmetry properties by Padilla et al. (6)] and a therapeutic p...