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Understanding of virus capsid organization and self-assembly mechanisms helps not only to get an insight into the protein interactions which render virus infectious, but also to advance new methods in nanotechnology which use capsid self-assembly to produce virus-like nanoparticles. As in abiotic nanostructures, the obstacles along this way are related not only to the nanoscopic size of capsids but also to their unconventional topology and symmetry. We have shown that proteins in the bovine papilloma virus (and other viruses of the exceptional papovavirus families) self-assemble into unprecedented chiral QC-like structures, with chiral pentagonal order in the faces and global dodecahedron geometry of the capsid. The corresponding dodecahedron net is commensurate with the chiral pentagonal quasilattice, with the asymmetric proteins put at the nodes of the quasilattice. We developed the nonlinear phason strain concept in the frame of the classical elasticity theory of QCs. The resulting approach allowed us to calculate the protein positions and thus to explain the protein organization in papovaviruses, in spite of its extreme complexity.