Hexagonal Inflation Tilings and Planar Monotiles

Abstract

Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focused on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces.

Taylor–Socolar Hexagonal Tilings as Model Sets

“…We have already pointed out different ways in [12] and Corollary 6.2. There are also other ways pointed out in [11]. The first is through the almost everywhere one-to-one mapping from the minimal dynamical hull of the Taylor-Socolar tilings to the minimal dynamical hull of half-hex tilings.…”

“…Nonetheless both tiling hulls have the half-hex hull and Q as factors, and amazingly both have the same dynamical zeta functions. A report on this work appears in this same volume [11] of Symmetry, where the Penrose tiling and the Taylor tilings are carefully compared. The approach there is based on the construction of the tilings through inflations, and the complications of the singular points of the hull arise from observing the special symmetries possessed by certain of the tilings.…”

“…Here we shall stick primarily to X Q since the corresponding results for X are easily inferred. See also [11] the structure of the hull X is determined (in a totally different way). The 2-adic group Q is then replaced by the two dimensional dyadic solenoid.…”

On the Penrose and Taylor–Socolar hexagonal tilings

“…Although it is shown in [2] that the two tiling spaces generated by T-S tilings and Penrose tilings define distinct MLD (mutual local derivabillity) classes, it is clear by now that the two types of tilings are intimately related, and indeed, modulo the choice of a coset, there is a mutual derivability. We can summarize some key points as follows: The process of producing double hexagon tiles from a pair (q, r) suggests that we might do it again, choosing a coset d + 3Q with d ≡ c mod 3P and then determining s ∈ Q 2 .…”

Tiling Spaces of Taylor-Socolar Tilings

“…3). The Penrose functional mono-tile tiling [3] arises from (1 + + 2 )-tiling [4,5]. This tiling is also based on hexagonal tiles as the TaylorSocolar tiling.…”