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Deltahedral views of fullerene polymorphism

Fullerenes and icosahedral virus particles share the underlying geometry applied by Buckminster Fuller in his geodesic dome designs. The basic plan involves the construction of polyhedra from 12 pentagons together with some number of hexagons, or the symmetrically equivalent construction of triangular faceted surface lattices (deltahedra) with 12 five-fold vertices and some number of six-fold vertices. All the possible designs for icosahedral viruses built according to this plan were enumerated according to the triangulation number T = (h2+hk+k2 of icosadelta-hedra formed by folding equilateral triangular nets with lattice vectors of indices h, kconnecting neighbouring five-fold vertices. Lower symmetry deltahedra can be constructed in which the vectors connecting five-fold vertices are not all identical. Applying the pentagon isolation rule, the possible designs for fullerenes with more than 20 hexagonal facets can be defined by the set of vectors in the surface lattice net of the corresponding deltahedra. Surface lattice symmetry and geometrical relations among fullerene isomers can be displayed more directly in unfolded deltahedral nets than in projected views of the deltahedra or their hexagonally and pentagonally facted dual polyhedra.