  Herein, fundamentals of topology and symmetry breaking are used to understand
crystallization and geometrical frustration in topologically close-packed
structures. This frames solidification from a new perspective that is unique
from thermodynamic discussions. Crystallization is considered as developing
from undercooled liquids, in which orientational order is characterized by a
surface of a sphere in four-dimensions (quaternion) with the binary polyhedral
representation of the preferred orientational order of atomic clustering
inscribed on its surface. As a consequence of the dimensionality of the
quaternion orientational order parameter, crystallization is seen as occurring
in "restricted dimensions." Homotopy theory is used to classify all
topologically stable defects, and third homotopy group defect elements are
considered to be generalized vortices that are available in superfluid ordered
systems. This topological perspective approaches the liquid-to-crystalline
solid transition in three-dimensions from fundamental concepts of:
Bose-Einstein condensation, the Mermin-Wagner theorem and
Berezinskii-Kosterlitz-Thouless (BKT) topological-ordering transitions. In
doing so, in this article, concepts that apply to superfluidity in "restricted
dimensions" are generalized in order to consider the solidification of
crystalline solid states.
