  In this article, the solidification phase transition between isotropic
liquids and icosahedral quasicrystalline structures is classified by the
application of a quaternion orientational order parameter. We suggest that the
formation of the E8 lattice in 8-dimensions, from which three-dimensional
icosahedral quasicrystals derive by projection, occurs without dimensionally
restricting the degrees of freedom of the quaternion orientational order
parameter to the quaternion plane. This is different from the development of
regular crystalline solids in three-dimensions, which occurs in "restricted
dimensions" and requires topological-ordering, as previously discussed by the
authors. In other words, herein, the formation of lattices in eight-dimensions
is 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 the
necessity of topological defects.
