  We introduce a tiling problem between bounded open convex polyforms
$\hat{P}\subset\mathbb{R}^2$ with directed and uniquely colored edges. If there
exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we show that one
can construct a monomorphism from the sandpile group
$G_{\Gamma_1}=\mathbb{Z}^{\Gamma_1}/\Delta(\mathbb{Z}^{\Gamma_1})$ on the
domain (graph) $\Gamma_1=\hat{P}_1\cap\mathbb{Z}^2$ to the respective group on
$\Gamma_2=\hat{P}_2\cap\mathbb{Z}^2$. We provide several examples of infinite
series of such tilings with polyforms converging to $\mathbb{R}^2$, and thus
the first definition of scaling-limits for the sandpile group on the plane.
Additional results include an exact sequence relating sandpile configurations
to harmonic functions, an alternative formula for the order of the sandpile
group based on a basis for the module of integer-valued harmonic functions, and
three examples of how to prove the existence of (cyclic) subgroups for infinite
families of sandpile groups by constructing appropriate integer-valued harmonic
functions. The main open question concerns if the scaling-limits of the
sandpile group for different sequences of polyforms converging to
$\mathbb{R}^2$ are isomorphic.
