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Figure 2.4
Geometric principles for generating icosahedral quasi-equivalent surface
lattices. These constructions show the relation between icosahedral symmetry axes
and quasi-equivalent symmetry axes. The latter are symmetry elements that hold only
in a local environment. (A) Hexamers are initially considered planar, and pentamers
are considered convex, introducing curvature in the sheet of hexamers when they are
inserted. The closed icosahedral shell, composed of hexamers and pentamers, is generated
by inserting 12 pentamers at appropriate positions in the hexamer net. To construct a
model of a particular quasi-equivalent lattice, one face of an icosahedron is generated in
the hexagon net. The origin is replaced by a pentamer, and the (h,k) hexamer is replaced
by a pentamer. The third replaced hexamer is identified by threefold symmetry (i.e.,
complete the equilateral triangle of the face). (b). Seven hexamer units (bold outlines in
(a)) defined by the
= 3 icosahedral face defined
in (a) has been shaded. A three-dimensional model of the lattice can be generated by
arranging 20 identical faces of the icosahedron as shown, and folded into a quasi-
equivalent icosahedron. (c) Cardboard models of several icosahedral quasi-equivalent
surface lattices constructed using the method described above. The procedure for
generating quasi-equivalent models described here does not exactly correspond to that
the described by Caspar & Klug (1962); however, the final models are identical with
those described in their paper. Reproduced with permission from Johnson, J. E., and
Speir, J. A. (1997) Quasi-equivalent viruses: a paradigm for protein assemblies,
T
= 3 lattice choice are shown, and the
T
J.
Mol. Biol
., 269(5), 665-675.
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