  We consider {\em L-graphs}, that is contact graphs of axis-aligned L-shapes
in the plane, all with the same rotation. We provide several characterizations
of L-graphs, drawing connections to Schnyder realizers and canonical orders of
maximally planar graphs. We show that every contact system of L's can always be
converted to an equivalent one with equilateral L's. This can be used to show a
stronger version of a result of Thomassen, namely, that every planar graph can
be represented as a contact system of square-based cuboids.
  We also study a slightly more restricted version of equilateral L-contact
systems and show that these are equivalent to homothetic triangle contact
representations of maximally planar graphs. We believe that this new
interpretation of the problem might allow for efficient algorithms to find
homothetic triangle contact representations, that do not use Schramm's monster
packing theorem.
