Orbits of crystallographic embedding of noncrystallographic groups and applications to virology
Abstract
The architecture of infinite structures with noncrystallographic symmetries can be modeled via aperiodic tilings, but a systematic construction method for finite structures with noncrystallographic symmetry at different radial levels is still lacking. We present here a group theoretical method for the construction of finite nested point set with noncrystallographic symmetry. Akin to the construction of quasicrystals, we embed a noncrystallographic group $G$ into the point group $\mathcal{P}$ of a higher dimensional lattice and construct the chains of all $G$containing subgroups. We determine the orbits of lattice points under such subgroups, and show that their projection into a lower dimensional $G$invariant subspace consists of nested point sets with $G$symmetry at each radial level. The number of different radial levels is bounded by the index of $G$ in the subgroup of $\mathcal{P}$. In the case of icosahedral symmetry, we determine all subgroup chains explicitly and illustrate that these point sets in projection provide blueprints that approximate the organisation of simple viral capsids, encoding information on the structural organisation of capsid proteins and the genomic material collectively, based on two case studies. Contrary to the affine extensions previously introduced, these orbits endow virus architecture with an underlying finite group structure, which lends itself better for the modelling of its dynamic properties than its infinite dimensional counterpart.
 Publication:

arXiv eprints
 Pub Date:
 November 2014
 arXiv:
 arXiv:1411.2115
 Bibcode:
 2014arXiv1411.2115T
 Keywords:

 Mathematical Physics;
 Mathematics  Group Theory;
 Mathematics  Metric Geometry
 EPrint:
 New version, title and contents changed. Accepted in Acta Crystallographica A