It is well known that sequences of crystals with Mackay icosahedral motif and increasing lattice parameters exist converging to the icosahedral quasi-crystal 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 degrees rotations around an irrational axis [tau 10] or an irrational angle of 44.48 degrees 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 Sigma = 3 and extending through Sigma = 7, 19, 49, 129, 337, ..., infinity created by rotation around [1 1 1] axis. It is also noted that the boundaries of higher CSL values (Sigma > 7) are composed of a combination of structural units from Sigma = 3 and Sigma = 7 boundaries.