It was recalled that such crystals were examples of long-range ordered aperiodic structures with non-crystallographic symmetry. Rational approximants were made up of the same motifs as the quasicrystal, but had a crystal-like repeated unit cell. Both of the structures could be described by projections from n-dimensional space onto 3-dimensional physical space. Coherent grain boundaries could be created by interfacing quasi-crystals or approximants of differing orientation. It was noted that the familiar concepts of coincidence site lattice, DSC, and O-lattice could be applied to such interfaces. It was shown that, for 180 rotations (twins) in icosahedral quasi-crystals, the coincidence ratio and the density of a given twin plane were related by 2 = constant, where  was equal to unity for odd twin axes and equal to 4 for even twin axes. This implied that even twin axes had lower densities of the twin plane for comparable values of the coincidence ratio. It was suggested that this explained why  = 4 was note observed experimentally for icosahedral crystals, while  = 5 often occurred.

O.Radulescu, D.H.Warrington: Materials Science Forum, 1996, 207-209, 329-32