  We consider the edge-isoperimetric problem on the graph of the infinite grid
$\mathbb{N}^{2}$ in the $\ell_{\infty}$ metric. We first show that the
solutions are not nested, so that techniques other than compressions have to be
used. We then show that for any given volume of sets in $\mathbb{N}^{2}$, there
exists an optimal set of a specific geometric form and describe this form. We
continue on to prove that the optimal perimeter has asymptotic growth rate
$2\sqrt{7x}$ as a function of the volume and obtain upper and lower bounds for
the optimal perimeter which are within the small additive constant of
$\frac{35}{2}$ of one another, thus effectively solving the discrete
isoperimetric inequality on this graph. Finally, we prove that there exist
arbitrarily long consecutive values of the volume for which the minimum
perimeter is the same.
