Virus capsids are made from building block proteins that are repeatedly synthesized from a few genes in the virus genome. In this article in Microbiology Today (pdf) Reidun Twarock and Tom Keef discuss the mathematics behind the remarkable symmetry of the viral protective coat and demonstrate how this is far from accidental:
The simplest viruses are nanometre-sized particles consisting of genomic RNA or DNA surrounded by protective protein containers. Like Trojan horses, these containers, or capsids, provide protection and transport for the viral genomes between hosts. One of their remarkable features is the wealth of well-ordered patterns in which capsid proteins are arranged in these containers. It makes you wonder… is there more to this display of beauty than mere aesthetics?