  By regular tessellation, we mean any hyperbolic 3-manifold tessellated by
ideal Platonic solids such that the symmetry group acts transitively on
oriented flags. A regular tessellation has an invariant we call the cusp
modulus. For small cusp modulus, we classify all regular tessellations. For
large cusp modulus, we prove that a regular tessellations has to be infinite
volume if its fundamental group is generated by peripheral curves only. This
shows that there are at least 19 and at most 21 link complements that are
regular tessellations (computer experiments suggest that at least one of the
two remaining cases likely fails to be a link complement, but so far we have no
proof). In particular, we complete the classification of all principal
congruence link complements given in Baker and Reid for the cases of
discriminant D=-3 and D=-4. We only describe the manifolds arising as
complements of links here with a future publication "Regular Tessellation
Links" giving explicit pictures of these links.
