# An aperiodic monotile that forces nonperiodicity through dendrites

## TL;DR

This paper presents a new aperiodic hexagonal monotile that enforces nonperiodic tilings through dendritic growth rules, introducing a novel method for generating nonperiodic plane tilings.

## Contribution

The authors introduce a new aperiodic monotile with a local growth rule based on dendrites, expanding the methods for creating nonperiodic tilings.

## Key findings

- The monotile admits infinitely many tilings without translational symmetry.
- The dendritic growth rule enforces nonperiodicity in tilings.
- The method provides a new approach to generate aperiodic tilings.

## Abstract

We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar--Taylor monotile, but can be realised by shape alone. The second is a local growth rule; a direct isometry of our monotile can be added to any patch of tiles provided that a tree on the monotile connects continuously with a tree on one of its neighbouring tiles. This condition forces tilings to grow along dendrites, which ultimately results in nonperiodic tilings. Our local growth rule initiates a new method to produce tilings of the plane.