  We review the Markov theoretic description of 1D aperiodic crystals,
describing the stacking-faulted crystal polytype as a special case of an
aperiodic crystal. Under this description we generalise the centrosymmetric
unit cell underlying a topologically centrosymmetric crystal to a reversible
Markov chain underlying a reversible aperiodic crystal. We show that for the
close-packed structure, almost all stackings are irreversible when the
interaction reichweite is greater than 4. Moreover, we present an analytic
expression of the scattering cross section of a large class of stacking
disordered aperiodic crystals, lacking translational symmetry of their layers,
including ice and opaline silica (opal CT). We then relate the observed
stackings and their underlying reichweite to the physics of various nucleation
and growth processes of disordered ice.
