  Given a lattice $L$ in the plane, we define the affiliated deep hole lattice
$H(L)$ to be spanned by a shortest vector of $L$ and a deep hole of $L$
contained in the triangle with sides corresponding to the shortest basis
vectors. We study the geometric and arithmetic properties of deep hole
lattices. In particular we investigate conditions on $L$ under which $H(L)$ is
well-rounded and prove that $H(L)$ is defined over the same field as $L$. For
the period lattice corresponding to an isomorphism class of elliptic curves, we
produce a finite sequence of deep hole lattices ending with a well-rounded
lattice which corresponds to a point on the boundary arc of the fundamental
strip under the action of $\operatorname{SL}_2(\mathbb{Z})$ on the upper
halfplane. In the case of CM elliptic curves, we prove that all elliptic curves
generated by this sequence are isogenous to each other and produce bounds on
the degree of isogeny. Finally, we produce a counting estimate for the planar
lattices with a prescribed deep hole lattice.
