Crystallization of the undertile

Abstract

We classify all periodic tilings with a monotile known as $\mathbf{R1}$ and called here the undertile, which underlies several aperiodic monotiles such as the Taylor-Socolar one. The classification is done using deflation, through a study of fracture lines between zones where deflation can be performed. A full classification of fracture patterns is given, that also applies to non-periodic tilings with the undertile. This can be used to reprove non-periodicity of some existing monotiles based on the undertile.