A toolkit for the construction of icosahedral particles with local   symmetry axes
  A formalism is developed which allows to determine the locations of all local symmetry axes of three-dimensional particles with overall icosahedral symmetry. It relies on the fact that the root system of the non-crystallographic Coxeter group H_3 encodes the locations of the planes of reflection that generate the discrete rotational symmetries of the particles. Via an appropriate extension of the root system, new planes of reflection are introduced which determine local axes of rotational symmetry. An easy-to-implement formalism is derived that allows to compute the surface structure of any three-dimensional icosahedral particle with local symmetries. It can be used also for particles with overall octahedral and tetrahedral symmetry in conjunction with the root systems of the corresponding reflection groups.   Applications to viruses are discussed explicitly. It is shown that the concept of quasi-equivalence in Caspar-Klug Theory corresponds to the special case of local six-fold symmetry axes contained in the theory developed here, and the corresponding geometries can hence be obtained with this formalism based on the root system of H_3.   Moreover, as a by-product, the theory answers the long-standing open question why only certain types of capsomeres, i.e. clusters of protein subunits, are observed in the surface structures of viruses. Since the types of the capsomeres are determined by the orders of the local symmetry axes on which they are located, the possible types of capsomeres are restricted by the spectrum of local symmetry axes allowed by the theory. Based on this we determine the spectrum of all capsomere types that may occur in viral capsids and give explicit examples for the lower-order cases. 
Reidun Twarock