# An aperiodic tile with edge-to-edge orientational matching rules

## TL;DR

This paper introduces a single hexagonal tile with novel edge-to-edge matching rules that enforce non-periodic tilings, providing a new approach to aperiodicity with rigorous mathematical properties.

## Contribution

The authors present a new aperiodic tile with unique edge matching rules, including a novel charge-based rule, and prove its aperiodicity and dynamical properties.

## Key findings

- Tile enforces non-periodic tilings with simple rules
- Hull of tilings is uniquely ergodic and minimal
- Tilings have pure point spectrum and model set structure

## Abstract

We present a single, connected tile which can tile the plane but only non-periodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar--Taylor tilings. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity. We show that the hull of all tilings satisfying our rules is uniquely ergodic and that almost all tilings in the hull belong to a minimal core of tilings generated by substitution. Identifying tilings which are charge-flips of each other, these tilings are shown to have pure point dynamical spectrum and a regular model set structure.