# Shape Convergence for Aggregate Tiles in Conformal Tilings

## TL;DR

This paper proves that in conformal tilings derived from a substitution tiling, the shapes of aggregate tiles converge to their Euclidean counterparts as the subdivision process progresses infinitely.

## Contribution

It establishes the shape convergence of aggregate tiles in conformal tilings associated with substitution tilings, a novel result in tiling theory.

## Key findings

- Aggregate tiles converge in shape as subdivision increases
- Conformal tilings approximate Euclidean tiles in the limit
- Provides a rigorous proof of shape convergence in tiling transformations

## Abstract

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 $T$.