Coincidence-site lattices as rational approximants to irrational twins

It is well known that sequences of crystals with Mackay icosahedral motif and increasing lattice parameters exist converging to the icosahedral quasicrystal in the limit. They are known as rational approximants. It has also been demonstrated that it is possible to create icosahedral symmetry by irrational twins involving five variants by 72 rotations around an irrational axis [s 1 0] or an irrational angle of 44.48 around a rotation axis [1 1 1]. These twinned crystals do not share a coincidence site lattice. In this paper, it is demonstrated that the above twinning relationship arises in the limit of a sequence of coincidence site lattices starting with the cubic twins with S = 3 and extending through S = 7, 19, 49, 129, 337, …, ¥ created by rotation around [1 1 1] axis. It is also noted that the boundaries of higher CSL values (S > 7) are composed of a combination of structural units from S = 3 and S = 7 boundaries