Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

Realizations of the abstract regular H 3 polyhedra

Regular polyhedra and related structures such as complexes and nets play a prominent role in the study of materials such as crystals, nanotubes and viruses. An abstract regular polyhedron {\cal P} is the combinatorial analog of a classical regular geometric polyhedron. It is a partially ordered set of elements called faces that are completely characterized by a string C-group (G, T), which consists of a group G generated by a set T of involutions. A realization R is a mapping from {\cal P} to a Euclidean G space that is compatible with the associated real orthogonal representation of G. This work discusses an approach to the theory of realizations of abstract regular polyhedra with an emphasis on the construction of a realization and its decomposition as a blend of subrealizations. To demonstrate the approach, it is applied to the polyhedra whose automorphism groups are abstractly isomorphic to the non-crystallographic Coxeter group H 3.

Structural puzzles in virology solved with an overarching icosahedral design principle

Viruses have evolved protein containers with a wide spectrum of icosahedral architectures to protect their genetic material. The geometric constraints defining these container designs, and their implications for viral evolution, are open problems in virology. The principle of quasi-equivalence is currently used to predict virus architecture, but improved imaging techniques have revealed increasing numbers of viral outliers. We show that this theory is a special case of an overarching design principle for icosahedral, as well as octahedral, architectures that can be formulated in terms of the Archimedean lattices and their duals. These surface structures encompass different blueprints for capsids with the same number of structural proteins, as well as for capsid architectures formed from a combination of minor and major capsid proteins, and are recurrent within viral lineages. They also apply to other icosahedral structures in nature, and offer alternative designs for man-made materials and nanocontainers in bionanotechnology.