Triangulated icosahedral lattices. **(A)** The hexagonal net used for constructing an icosahedron in also serves to illustrate the formation of equilateral facets in the triangulated icosahedral lattices. The facet of the triangulated lattice of an icosahedron of a given T number is specified by a vector **T** (H, K) from the origin (**O**). The unit vector **T** (1, 0) (black) in this representation, specifies the facet of the basic T=1 icosahedron. Other vectors such as **T** (1, 1), **T** (2, 0) and **T** (1, 2), for example, form the basis for successively larger T=3 (red), T=4 (green) and T=7 (blue) icosahedral facets, respectively. To construct a triangulated icosahedral lattice of a particular T number (= H^{2} +HK +K^{2}), its facet, which is an equilateral triangle, is generated, as specified by the vector **T** (H, K), using the underlying planar hexagonal net. Nineteen other triangles are then generated to give a template, as in , for constructing an icosahedron with a given T number. The black, red, green, and blue triangles, represent facets for the T=1, 3, 4 and 7 icosahedra, respectively. Each vertex of the facet triangle becomes a 5-fold when the template is folded into an icosahedron as shown in .

**(B)** The folded T=1 (H=1, K=0) and T=4 (H=2, K=0) templates, in black and green, respectively are shown for comparison. The subunit organization in these icosahedra is shown using ‘,’ to represent a “subunit” as in . In contrast to the T=1 icosahedron, which has 20 unit triangles, specified by the vector **T** (1, 0), with 60 subunits, the T=4 lattice, specified by **T** (2, 0), consists of 80 (20T) unit triangles with 240 (60T) subunits. The folded T=4 icosahedron clearly illustrates how the non-vertex 6-fold positions in the net remain hexavalent (indicated by hexagon symbols) whereas the vertex 6-fold positions in the net become pentavalent (indicated by pentagon symbol) upon folding. As in any triangulated icosahedron, with the exception of the unit triangle at the icosahedral 3-fold position, the other triangles exhibit local (or quasi) 3-fold symmetry; similarly, the adjacent unit triangles are related by local (or quasi) 2-fold axes, unless the midpoint of the edge coincides with icosahedral 2-fold axis. Each facet in the T=4 icosahedron is comprised of four triangles. The icosahedral 3-fold axis, at the center of the facet, divides the facet into three equivalent parts. Each part, consisting of one third of the central triangle and one of the remaining three triangles in the facet constitutes the icosahedral asymmetric unit. The four “subunits” in the asymmetric unit, one from the central triangle, and three from one of the other triangles in facet are shown in a different colors to indicate their quasi-equivalent environment. The subunit in the central triangle is colored in black, while the subunits in the other triangle are colored in red, blue and green. Application of the strict 5-, 3- and 2-fold rotations to these quasi-equivalent subunits in the asymmetric unit generates rest of the 240 subunits. The arrangement of the subunits clearly illustrate how the triangulation leads to the formation of the rings of 5- (red subunits) and 6- (black, green and blue around the hexavalent points) because of the triangulation. Furthermore, it also illustrates that although the formation of rings of 5- and 6 in a triangulated lattice is a geometrical necessity, clustering of the subunits into pentamers and hexamers is not obligatory. In the T=4 icosahedron shown here, either each subunit can be considered separately, or as a cluster of three subunits (trimer). With a trimer as the building block, one trimer (three black subunits) occupies the central triangle of the facet, at the icosahedral 3-fold position, and another “quasi-equivalent” trimer (red, green, and blue subunits) occupies the other triangle in the facet. Such a T=4 organization with trimers is observed in the case of alphaviruses (see ). See , and for folded T=3 (red facet) and T=7 (blue facet) icosahedral lattices, respectively.

## PubMed Commons