  A set of geometric graphs is {\em geometric-packable} if it can be
asymptotically packed into every sequence of drawings of the complete graph
$K_n$. For example, the set of geometric triangles is geometric-packable due to
the existence of Steiner Triple Systems. When $G$ is the $4$-cycle (or
$4$-cycle with a chord), we show that the set of plane drawings of $G$ is
geometric-packable. In contrast, the analogous statement is false when $G$ is
nearly any other planar Hamiltonian graph (with at most 3 possible exceptions).
A convex geometric graph is {\em convex-packable} if it can be asymptotically
packed into the convex drawings of the complete graphs. For each planar
Hamiltonian graph $G$, we determine whether or not a plane $G$ is
convex-packable. Many of our proofs explicitly construct these packings; in
these cases, the packings exhibit a symmetry that mirrors the vertex
transitivity of $K_n$.
