Janner A
Theoretical Physics, Radboud University Nijmegen, Toernooiveld 1, NL-6525 ED Nijmegen, The Netherlands.
Acta Crystallogr A. 2006 Sep;62(Pt 5):319-30. doi: 10.1107/S0108767306022227. Epub 2006 Aug 23.
The standard Caspar & Klug classification of icosahedral viruses by means of triangulation numbers and the more recent novel characterization of Twarock leading to a Penrose-like tessellation of the capsid of viruses not obeying the Caspar-Klug rules can be obtained as a special case in a new approach to the morphology of icosahedral viruses. Considered are polyhedra with icosahedral symmetry and rational indices. The law of rational indices, fundamental for crystals, implies vertices at points of a lattice (here icosahedral). In the present approach, in addition to the rotations of the icosahedral group 235, crystallographic scalings play an important rôle. Crystallographic means that the scalings leave the icosahedral lattice invariant or transform it to a sublattice (or to a superlattice). The combination of the rotations with these scalings (linear, planar and radial) permits edge, face and vertex decoration of the polyhedra. In the last case, satellite polyhedra are attached to the vertices of a central polyhedron, the whole being generated by the icosahedral group from a finite set of points with integer indices. Three viruses with a polyhedral enclosing form given by an icosahedron, a dodecahedron and a triacontahedron, respectively, are presented as illustration. Their cores share the same polyhedron as the capsid, both being in a crystallographic scaling relation.
二十面体病毒的标准卡斯帕 - 克卢格分类法（通过三角剖分数）以及特瓦罗克最近提出的新特征描述（导致不遵循卡斯帕 - 克卢格规则的病毒衣壳呈现类似彭罗斯镶嵌的结构），都可以在一种研究二十面体病毒形态的新方法中作为特殊情况得到。这里考虑的是具有二十面体对称性和有理指数的多面体。对晶体至关重要的有理指数定律意味着顶点位于晶格（这里是二十面体晶格）的点上。在当前方法中，除了二十面体群235的旋转外，晶体学缩放也起着重要作用。晶体学意味着这些缩放使二十面体晶格不变或将其变换为子晶格（或超晶格）。旋转与这些缩放（线性、平面和径向）的组合允许对多面体进行边、面和顶点装饰。在最后一种情况下，卫星多面体附着在中心多面体的顶点上，整个结构由二十面体群从一组具有整数指数的有限点生成。作为示例，展示了分别由二十面体、十二面体和三十面体给出多面体包封形式的三种病毒。它们的核心与衣壳共享相同的多面体，两者处于晶体学缩放关系。
