# Bridges 2010 Regular Paper

On Growth, Form and Yin-Yang

Michael Longuet-Higgins

(Proceedings pages 323–328)

## Abstract

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.

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