  Aperiodic tilings with a small number of prototiles are of particular
interest, both theoretically and for applications in crystallography. In this
direction, many people have tried to construct aperiodic tilings that are built
from a single prototile with nearest neighbour matching rules, which is then
called a monotile. One strand of the search for a planar monotile has focussed
on hexagonal analogues of Wang tiles. This led to two inflation tilings with
interesting structural details. Both possess aperiodic local rules that define
hulls with a model set structure. We review them in comparison, and clarify
their relation with the classic half-hex tiling. In particular, we formulate
various known results in a more comparative way, and augment them with some new
results on the geometry and the topology of the underlying tiling spaces.
