A self-similar aperiodic set of 19 Wang tiles

TL;DR

This paper introduces a new 19-tile Wang set that is self-similar, aperiodic, and minimal, expanding the known examples of such sets and demonstrating their properties through a unique composition and morphism approach.

Contribution

It presents the second smallest known self-similar aperiodic Wang tile set and proves its properties using a novel recognizability and composition framework.

Findings

The set $al{U}$ is self-similar and aperiodic.

Existence of an expansive, primitive, recognizable 2D morphism.

The set is minimal and comparable in size to Ammann's set of 16 tiles.

Abstract

We define a Wang tile set $\mathcal{U}$ of cardinality 19 and show that the set $\Omega_\mathcal{U}$ of all valid Wang tilings $\mathbb{Z}^2\to\mathcal{U}$ is self-similar, aperiodic and is a minimal subshift of $\mathcal{U}^{\mathbb{Z}^2}$. Thus $\mathcal{U}$ is the second smallest self-similar aperiodic Wang tile set known after Ammann's set of 16 Wang tiles. The proof is based on the unique composition property. We prove the existence of an expansive, primitive and recognizable $2$-dimensional morphism $\omega:\Omega_\mathcal{U}\to\Omega_\mathcal{U}$ that is onto up to a shift. The proof of recognizability is done in two steps using at each step the same criteria (the existence of marker tiles) for proving the existence of a recognizable one-dimensional substitution that sends each tile either on a single tile or on a domino of two tiles.