A construction of the hat tilings by a Markov partition

TL;DR

This paper introduces a straightforward method for constructing hat tilings using a superimposed triangular grid on a colored image, establishing a correspondence between the construction and valid tilings.

Contribution

It provides a simple, geometric construction method for hat tilings and proves its completeness in generating all valid tilings.

Findings

The construction produces valid hat tilings.

Every valid hat tiling can be obtained through this method.

Abstract

We present a simple construction of hat tilings. The construction can be carried out by superimposing a triangular grid on a specially colored image and reading off the orientation of the tiles. We show that our construction produces valid hat tilings, and conversely, in an appropriate sense that is made precise in the paper, that every valid hat tiling can be obtained in this way.