Experimental Results on Potential Markov Partitions for Wang Shifts

TL;DR

This paper explores experimental methods to identify potential Markov partitions for specific Wang tile sets, including aperiodic examples related to Penrose and Ammann tilings, and analyzes these partitions using recent theoretical tools.

Contribution

It introduces an experimental methodology for generating potential Markov partitions for Wang shifts and applies recent theoretical analysis to these partitions.

Findings

Identified potential Markov partitions for three Wang tile protosets.

Applied Labbé's theory to analyze experimentally discovered partitions.

Provided insights into encoding aperiodic tilings with Markov partitions.

Abstract

In this article we discuss potential Markov partitions for three different Wang tile protosets. The first partition is for the order-24 aperiodic Wang tile protoset that was recently shown in the Ph.D. thesis of H. Jang to encode all tilings by the Penrose rhombs. The second is a partition for an order-16 aperiodic Wang protoset that encodes all tilings by the Ammann A2 aperiodic protoset. The third partition is for an order-11 Wang tile protoset identified by Jeandel and Rao as a candidate order-11 aperiodic Wang tile protoset. The emphasis is on some experimental methodology to generate potential Markov partitions that encode tilings. We also apply some of the theory developed by Labb\'{e} in analyzing such an experimentally discovered partition.