  There are three kinds of solid states of matter that can exist in physical
space: quasicrystalline (quasiperiodic), crystalline (periodic) and amorphous
(aperiodic). Herein, we consider the degree of orientational order that
develops upon the formation of a solid state to be characterized by the
application of quaternion numbers. The formation of icosahedral
quasicrystalline solids is considered alongside the development of bulk
superfluidity, characterized by a complex order parameter, that occurs by
spontaneous symmetry breaking in three-dimensions. Crystalline solid states are
viewed as higher-dimensional analogues to phase-coherent topologically-ordered
superfluid states of matter that develop in restricted dimensions
(Hohenberg-Mermin- Wagner theorem). Lastly, amorphous solid states are viewed
as dual to crystalline solids, in analogy to Mott-insulating states of matter
that are dual to topologically-ordered superfluids.
