The architecture of infinite structures with non-crystallographic symmetries can be modelledviaaperiodic tilings, but a systematic construction method for finite structures with non-crystallographic symmetry at different radial levels is still lacking. This paper presents a group theoretical method for the construction of finite nested point sets with non-crystallographic symmetry. Akin to the construction of quasicrystals, a non-crystallographic groupGis embedded into the point group {\cal P} of a higher-dimensional lattice and the chains of allG-containing subgroups are constructed. The orbits of lattice points under such subgroups are determined, and it is shown that their projection into a lower-dimensionalG-invariant subspace consists of nested point sets withG-symmetry at each radial level. The number of different radial levels is bounded by the index ofGin the subgroup of {\cal P}. In the case of icosahedral symmetry, all subgroup chains are determined explicitly and it is illustrated that these point sets in projection provide blueprints that approximate the organization of simple viral capsids, encoding information on the structural organization 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 to the modelling of dynamic properties than its infinite-dimensional counterpart.
The Exterior Square of a Bieberbach Group with Quaternion Point Group of Order Eight
