Shape Convergence for Aggregate Tiles in Conformal Tilings

Given a substitution tiling $T$ of the plane with subdivision operator $\tau$, we study the conformal tilings $\mathcal{T}_n$ associated with $\tau^n T$. We prove that aggregate tiles within $\mathcal{T}_n$ converge in shape as $n\rightarrow \infty$ to their associated Euclidean tiles in $T$.