On Growth, Form and Yin-Yang

Michael Longuet-Higgins
Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (2010)
Pages 323–328


The outer coverings of many virus particles have the rotational symmetry of an icosahedron. How does this external sheath assemble itself? As a simple model we assume each unit or “capsomere” to be represented by a small sphere, attracted to the center of the virus. Geometrically this is similar to studying how a random set of equal spheres would behave if in contact with a given sphere of constant radius, or equivalently a set of N small, equal circles drawn randomly on a fixed sphere. We adopt a “yin-yang” method: first, we jostle the circles in a random manner (the yin phase), and then we allow them expand slightly (the yang phase). When N = 4, 6 or 24 the circles self-assemble to the pattern corresponding to a snub polyhedron, but when N = 60 the densest packing is irregular. When N = 72, as is found in the polyoma virus and other organisms, the packing does not become regular unless the circles are first assembled into a “flower”. Each flower has five circles or “petals” surrounding a central circle. When subjected to yin-yang the 12 flowers converge into the observed form. It is inferred that the virus sheath is assembled in this manner.