Experimental Results on Potential Markov Partitions for Wang Shifts

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 Labbé in analyzing such an experimentally discovered partition.

arXiv commentIn this version of the article, the focus shifted from proving that the partition for order-24 Wang tile protoset derived from the Penrose rhombs is a Markov partition to a more general experimental approach to finding potential Markov partitions for aperiodic Wang shifts. The proposition in error in the previous version is removed in the update