An aperiodic monotile for the tiler

TL;DR

This paper introduces a new aperiodic monotile capable of tiling the plane without repeating patterns, expanding the understanding of aperiodic tilings and solving a longstanding mathematical problem.

Contribution

It presents a novel aperiodic monotile that does not require motifs for aperiodicity, building on previous work and providing a new proof of aperiodicity.

Findings

The monotile can tile the plane aperiodically without motifs.

The monotile is based on a two-layer design.

A proof of aperiodicity is provided.

Abstract

Can the entire plane be paved with a single tile that forces aperiodicity? This is known as the ein Stein problem (in German, ein Stein means one tile). This paper presents an aperiodic monotile for the tiler. It is based on the monotile developed by Taylor and Socolar (whose aperiodicity is forced by means of a non-connected tile that is mainly hexagonal) and motif-based hexagonal tilings that followed this major discovery. The proposed monotile consists of two layers. No motif is needed to make the monotile aperiodic. Additional motifs can be added to the monotile to provide some insights. The proof of aperiodicity is presented with the use of such motifs.