arXiv subjects

Casey Mann

Publications and source records attributed to Casey Mann.

3 recordsLinked to original sources

arXiv2023

Experimental Results on Potential Markov Partitions for Wang Shifts

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.

math.DS↗

arXiv2022

Nonexpansive directions in the Jeandel-Rao Wang shift

We show that $\{0,\varphi+3,-3\varphi+2,-\varphi+\frac{5}{2}\}$ is the set of slopes of nonexpansive directions for a minimal subshift in the Jeandel-Rao Wang shift, where $\varphi=(1+\sqrt{5})/2$ is the golden mean. This set is a topological invariant allowing to distinguish the Jeandel-Rao Wang shift from other subshifts. Moreover, we describe the combinatorial structure of the two resolutions of the Conway worms along the nonexpansive directions in terms of irrational rotations of the unit interval. The introduction finishes with pictures of nonperiodic Wang tilings corresponding to what Conway called the cartwheel tiling in the context of Penrose tilings. The article concludes with open questions regarding the description of octopods and essential holes in the Jeandel-Rao Wang shift.

math.DS↗

arXiv2015

Convex pentagons that admit $i$-block transitive tilings

The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. Previous to this article, there were 14 known distinct kinds of convex pentagons that admit tilings of the plane. Five of these types admit tile-transitive tilings (i.e. there is a single transitivity class with respect to the symmetry group of the tiling). The remaining 9 types do not admit tile-transitive tilings, but do admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters form tile-2-transitive or tile-3-transitive tilings. In this article, we present some combinatorial results concerning pentagons that admit $i$-block transitive tilings for $i \in \mathbb{N}$. These results form the basis for an automated approach to finding all pentagons that admit $i$-block transitive tilings for each $i \in \mathbb{N}$. We will present the methods of this algorithm and the results of the computer searches so far, which includes a complete classification of all pentagons admitting 1-, 2-, and 3-block transitive tilings, among which is a new 15th type of convex pentagon that admits a tile-3-transitive tiling.

math.MG↗