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In this article, the solidification phase transition between isotropicliquids and icosahedral quasicrystalline structures is classified by theapplication of a quaternion orientational order parameter. We suggest that theformation of the E8 lattice in 8-dimensions, from which three-dimensionalicosahedral quasicrystals derive by projection, occurs without dimensionallyrestricting the degrees of freedom of the quaternion orientational orderparameter to the quaternion plane. This is different from the development ofregular crystalline solids in three-dimensions, which occurs in "restricteddimensions" and requires topological-ordering, as previously discussed by theauthors. In other words, herein, the formation of lattices in eight-dimensionsis considered as a higher-dimensional analogy to the formation of "bulk"superfluidity. It follows that, the solidification of lattices in 8-dimensions(and derivative icosahedral quasicrystalline solids) should occur without thenecessity of topological defects.

Solidification of Icosahedral Quasicrystals: Viewpoint Based on a Quaternion Orientational Order Parameter and the Formation of the E8 Lattice