Corona limits of tilings : Periodic case

TL;DR

This paper investigates the asymptotic shape of growing regions in periodic tilings, establishing that corona limits are uniquely determined by directional speeds and are convex polyhedra, with implications for crystal growth modeling.

Contribution

It provides a new proof that corona limits of periodic tilings are centrally symmetric convex polyhedra, characterizing them via directional speeds.

Findings

Corona limits are uniquely characterized by directional speeds.

Corona limits of periodic tilings are centrally symmetric convex polyhedra.

The paper offers an alternative proof of known geometric properties.

Abstract

We study the limit shape of successive coronas of a tiling, which models the growth of crystals. We define basic terminologies and discuss the existence and uniqueness of corona limits, and then prove that corona limits are completely characterized by directional speeds. As an application, we give another proof that the corona limit of a periodic tiling is a centrally symmetric convex polyhedron (see [Zhuravlev 2001], [Maleev-Shutov 2011]).