Experimental Results on Potential Markov Partitions for Wang Shifts

A tiling of the Euclidean plane $\mathbb{E}^{2}$ is a collection $\{T_{i}\}$ of distinct sets called tiles (which are typically topological disks) such that $\cup T_{i}=\mathbb{E}^{2}$ and for all $i\neq j$, $\text{int}(T_{i})\cap\text{int}(T_{j})=\emptyset$. A protoset for a tiling $\mathscr{T}$ is a collection of tiles $\mathcal{T}$ such that each tile of $\mathscr{T}$ is congruent to a tile in $\mathcal{T}$, in which case we say that $\mathcal{T}$ admits the tiling $\mathscr{T}$. The symmetry group of a tiling $\mathscr{T}$ is the set of rigid planar transformations $\mathcal{S}(\mathscr{T})$ such that $\sigma(\mathscr{T})=\mathscr{T}$ for all $\sigma\in\mathcal{S}(\mathscr{T})$. If $\sigma(\mathscr{T})$ contains two nonparallel translations, we say that $\mathscr{T}$ is periodic; otherwise, we say $\mathscr{T}$ is nonperiodic. If $\mathcal{T}$ admits at least one tiling of the plane and every such tiling is nonperiodic, we say that $\mathcal{T}$ is an aperiodic protoset.

Aperiodic protosets are somewhat rare, and indeed, until the 1960s, no aperiodic protosets were known to exist. H. Wang famously conjectured that aperiodic protosets cannot exist [Wan61]; that is, Wang conjectured that any protoset that admits a tiling of the plane must admit at least one periodic tiling. Wang’s conjecture was refuted in 1964 by his doctoral student, R. Berger, in his PhD thesis [Ber65, Ber66] where an aperiodic protoset consisting of over 20,000 edge-marked squares called Wang tiles was described. Wang tiles are a bit of a special case in the theory of tilings in that only translations are allowed in forming tilings from Wang tile protosets; if rotations and reflections are also allowed, then any single Wang tile admits periodic tilings of the plane, and so without this restriction the problem is trivial. Since Berger’s original discovery, several lower-order aperiodic Wang tile protosets have been discovered, culminating in 2015 with the discovery by E. Jeandel and M. Rao [JR21] of an order-11 aperiodic Wang tile protoset. In this same work the authors proved that 11 is the minimum possible order for an aperiodic Wang tile protoset. The aperiodic Jeandel-Rao protoset is seen in Figures 1, and here we are using the same labeling for the Jeandel-Rao protoset as given by Labbé in [Lab21].

Aperiodic protosets of tiles other than Wang tiles are also known. In 1971, R. Robinson developed a way to encode the idea of Berger’s original aperiodic protoset into an order-6 aperiodic protoset consisting of shapes with notched edges and modified corners (Figure 2(a)). In 1974, R. Penrose discovered an order-6 aperiodic protoset that he was later able to modify to produce an order-2 aperiodic protoset. There are a few varieties of the order-2 aperiodic Penrose protoset, one of which, called P3 or the Penrose rhombs, is depicted in Figure 2(b); this protoset will be discussed further this article. More recently, in 2023 two singleton aperiodic protosets was discovered ([Smi+23a, Smi+23]) - these “aperiodic monotiles” are called the hat (Figure 2(c)) and spectre (Figure 2(d)); we note that no edge-matching rules are needed to enforce nonperiodicity with the hat and spectre, unlike the Penrose rhombs, for example.

Let $\mathcal{T}_{0}=\{T_{0},T_{1},T_{2},\ldots,T_{10}\}$ be the aperiodic Jeandel-Rao Wang tile protoset (Figure 1). The set of all tilings admitted by $\mathcal{T}_{0}$ is called the Jeandel-Rao Wang shift and is denoted $\Omega_{0}$. In [Lab21], S. Labbé presented a special partition $\mathcal{P}_{0}=\{P_{0},P_{1},\ldots,P_{10}\}$ (Figure 3) of the two-dimensional torus $\mathbf{T}$, along with a $\mathbb{Z}^{2}$ action on $\mathbf{T}$, that together give rise tilings in $\Omega_{0}$. Specifically, let $\varphi=(1+\sqrt{5})/2$ be the golden mean and let $\Gamma_{0}$ be the lattice in $\mathbb{R}^{2}$ generated by $(\varphi,0)$ and $(1,\varphi+3)$. Then the partitioned torus is $\mathbf{T}=\mathbb{R}^{2}/\Gamma_{0}$ and the $\mathbb{Z}^{2}$ action $R_{0}$ on $\mathbf{T}$ is defined by $R_{0}^{\mathbf{n}}(\boldsymbol{x})=\boldsymbol{x}+\mathbf{n}\pmod{\Gamma_{0}}$.

Here we describe how the partition $\mathcal{P}_{0}$ of $\mathbf{T}$ and the action $R_{0}$ encode tilings by $\mathcal{T}_{0}$. Let $p\in\mathbf{T}$ and consider the orbit $\mathcal{O}_{\mathbb{R}_{0}}(p)=\{R_{0}^{\mathbf{n}}(p):\mathbf{n}\in\mathbb{Z}^{2}\}$. Then $\mathcal{O}_{R_{0}}(p)$ corresponds to a tiling in $\Omega_{0}$ in the following way:

This idea is depicted in Figure 3. There are several technical details that Labbé proved so that the correspondence between orbits of points in the partitioned torus correspond to tilings in an unambiguous way, but this is the gist of it. It is interesting - even somewhat amazing - that most tilings admitted by $\mathcal{T}_{0}$ are encoded by $\mathcal{P}_{0}$ and $R_{0}$!

In this article, we present some partitions, discovered experimentally, that are analogous to the Markov partition $\mathcal{P}_{0}$ for the Jeandel-Rao protoset, but for other Wang tile protosets. Further, we discuss how to identify other potential Markov partitions for Wang shifts in which tilings exhibit “golden rotational” behavior similar to $R_{0}$ with $\mathcal{P}_{0}$. We will also discuss theoretical framework described by Labbé in [Lab21] and apply it to one of these new partitions in an effort to describe the full space of tilings arising from the partition.

Geometrically, a Wang tile is a square with labeled (or colored) sides that serve as edge matching rules; two Wang tiles may meet along an edge if corresponding labels match. In forming a Wang tiling from copies of tiles in a protoset of Wang tiles, we allow only translates of the prototiles to be used, and we require that the tiles to meet edge-to-edge, respecting the matching rules. The edge-to-edge matching rules and restriction to translations can always be enforced geometrically, as in Figure 1. Wang tilings can be aligned with the integer lattice $\mathbb{Z}^{2}$ so that each tile has its lower left-hand corner in $\mathbb{Z}^{2}$, and thus we may view a Wang tiling as a decoration of $\mathbb{Z}^{2}$ such that each point of $\mathbb{Z}^{2}$ is “colored” with a single tile.

More formally, a Wang tile can be represented as a tuple of four colors $(r,t,\ell,b)\in V\times H\times V\times H$ where $V$ and $H$ are finite sets (the vertical and horizontal colors, respectively). For any Wang tile $T=(r,t,\ell,b)$, let $\text{Right}(T)=r$, $\text{Top}(T)=t$, $\text{Left}(T)=\ell$, and $\text{Bottom}(T)=b$ be the colors of the right, top, left, and right sides of $\tau$, respectively. Let $\mathcal{T}=\{T_{0},T_{1},\ldots,T_{n-1}\}$ be a protoset of Wang tiles. In forming a tiling with the protoset $\mathcal{T}$, we will think of the prototiles of $\mathcal{T}$ as symbols used to label the integer lattice $\mathbb{Z}^{2}$. This motivates us to define

to be the configuration space of $\mathcal{T}$, and each element $x\in\mathcal{T}^{\mathbb{Z}^{2}}$ is called a configuration. We will use the notation $x_{\mathbf{n}}=x(\mathbf{n})$ to indicate the value of $x$ at $\mathbf{n}\in\mathbb{Z}^{2}$ so that if $x_{\mathbf{n}}=T_{i}$ we understand that tile $T_{i}$ is located at position $\mathbf{n}$ (i.e,, oriented so that its corners lie in $\mathbb{Z}^{2}$ and the lower left-hand corner is at $\mathbf{n}$). It is possible that a configuration in $\mathcal{T}^{\mathbb{Z}^{2}}$ does not represent a valid tiling; so far, this construction does not account for the matching rules of the tiles. To account for the matching rules, we define a configuration $x$ to be valid if for every $\mathbf{n}\in\mathbb{Z}^{2}$, we have $\text{Right}(x_{\mathbf{n}})=\text{Left}(x_{\textbf{n}+(1,0)})$ and $\text{Top}(x_{\mathbf{n}})=\text{Bottom}(x_{\textbf{n}+(0,1)})$. The Wang shift of $\mathcal{T}$ is the set $\Omega_{\mathcal{T}}$ of all valid configurations in $\mathcal{T}^{\mathbb{Z}^{2}}$. Thus, $\Omega_{\mathcal{T}}$ captures in a symbolic way the set of all valid Wang tilings admitted by the Wang tile protoset $\mathcal{T}$.

A topological dynamical system consists of a topological space $X$ and a group $G$ that acts on $X$ via a continuous function $S\!:\!:G\times X\rightarrow X$, and is denoted by a triple $(X,G,S)$. For fixed $g\in G$, let $S^{g}\!:\!X\rightarrow X$ be defined by $S^{g}(x)=S(g,x)$. The orbit of a point $x\in X$ under the action of $S$ on $G$ is the set $\mathcal{O}_{S}(x,G)=\{S^{g}(x):g\in G\}$, and when the group $G$ is understood by context, we will shorten this notation to just $\mathcal{O}_{S}(x)$. We say $S$ acts freely on $X$ if for all $x\in X$, $S^{g}(x)=x$ if and only if $g=e$ where $e\in G$ is the group identity element.

Let $\sigma:\mathbb{Z}^{2}\times\mathcal{T}^{\mathbb{Z}^{2}}\rightarrow\mathcal{T}^{\mathbb{Z}^{2}}$ be defined by $\sigma^{\mathbf{n}}(x_{\mathbf{m}})=x_{\mathbf{m}+\mathbf{n}}$. We call $\sigma$ the $\mathbb{Z}^{2}$ shift action on $\mathcal{T}^{\mathbb{Z}^{2}}$; the terminology “shift action” is accurately descriptive here since, for each $\mathbf{n}\in\mathbb{Z}^{2}$ and $x\in\mathcal{T}^{\mathbb{Z}^{2}}$, $\sigma^{\mathbf{n}}(x)$ is a translation of $x$ by $\mathbf{n}$. By placing the appropriate metric on $\mathcal{T}^{\mathbb{Z}^{2}}$, it is not too hard to see that $\Omega_{\mathcal{T}}\subseteq\mathcal{T}^{\mathbb{Z}^{2}}$ is a topological space so that $(\Omega_{\mathcal{T}},\mathbb{Z}^{2},\sigma)$ satisfies the definition of being a topological dynamical system. Moreover, $(\Omega_{\mathcal{T}},\mathbb{Z}^{2},\sigma)$ is a special kind of topological dynamical system called a shift of finite type (see Appendix A).

If $\sigma$ acts freely on $\Omega_{\mathcal{T}}$, we say that $\Omega_{\mathcal{T}}$ is an aperiodic shift, and we say that a Wang protoset $\mathcal{T}$ is an aperiodic protoset if the corresponding Wang shift $\Omega_{\mathcal{T}}$ is nonempty and aperiodic. Notice that this definition for aperiodic protoset agrees with with more general definition given in Section 1.1 if we restrict the allowable symmetries to translations only.

Here we will provide the broad strokes of Labbé’s [Lab21] general approach and construction of the dynamical system $(\mathcal{X}_{\mathcal{P},R},\mathbb{Z}^{2},\sigma)$ that is based on a partition $\mathcal{P}$ of a compact topological space $\mathbf{T}$ and a $\mathbb{Z}^{2}$ action $R$ on $\mathbf{T}$. The space $\mathcal{X}_{\mathcal{P},R}$ is the set of tilings encoded by the partition $\mathcal{P}$ and the action $R$, as exemplified by Figure 3. We will relegate much of the more technical details to Appendix A. The main point is that methodologies are described in [Lab21] that can be used to prove when $\mathcal{P}$ and $R$ produce a space $\mathcal{X}_{\mathcal{P},R}$ that is reasonable subspace (or subshift) of the space of all valid tilings $\Omega_{\mathcal{T}}$ admitted by $\mathcal{T}$.

First, let $\mathbf{T}$ be a compact metric space and let $I_{n}=\{0,1,\ldots,n-1\}$. A collection $\mathcal{P}=\{P_{i}\}_{i\in I_{n}}$ of disjoint open sets with $\mathbf{T}=\cup_{i\in I_{n}}\overline{P}_{i}$ is called a topological partition of $\mathbf{T}$, and the sets $P_{i}\in\mathcal{P}$ are called the atoms of the partition. Let $\mathcal{P}=\{P_{i}\}_{i\in I_{n}}$ is a topological partition of $\mathbf{T}$, and let $\mathcal{T}=\{T_{i}\}_{i\in I_{n}}$ be a protoset of Wang tiles having the same indexing set $I_{n}$ as $\mathcal{P}$. Let $R$ be a $\mathbb{Z}^{2}$ action on $\mathbf{T}$. Then $(\mathbf{T},\mathbb{Z}^{2},\mathbb{R})$ is a topological dynamical system, and with $\mathbf{T}$ so partitioned by $\mathcal{P}$, the hope is that the orbit of a point $p\in\mathbf{T}$ will “spell out” a tiling in $\Omega_{\mathcal{T}}$. More precisely, for each $p\in\mathbf{T}$, we can consider the orbit $\mathcal{O}_{R}(p)=\{R^{\mathbf{n}}(p):\mathbf{n}\in\mathbb{Z}^{2}\}$, and for each $\mathbf{n}\in\mathbb{Z}^{2}$, define the configuration $x\!:\!\mathbb{Z}^{2}\rightarrow\mathcal{T}$ in $\mathcal{T}^{\mathbb{Z}^{2}}$ by

The configuration $x$ may not be not well defined because for some $\mathbf{n}\in\mathbb{Z}^{2}$, the point $R^{\mathbf{n}}(p)$ may lie on the boundary of more than one atom of $P_{i}$. However, Labbé [Lab21] was able prove that this ambiguity can be removed in the following way: By specifying a direction $\mathbf{v}$ that is not parallel to any of the sides of the atoms of the partition, then if a point $a\in\mathcal{O}_{R}(p)$ falls on the boundary of an atom, the choice of tile associated with that point is the one in the direction of $\mathbf{v}$ from $a$. In this way, each point $p\in\mathbf{T}$ can be associated uniquely with a configuration $x\in\mathcal{T}^{\mathbb{Z}^{2}}$, thereby defining a mapping $\text{SR}^{\mathbf{v}}\!:\!\mathbf{T}\rightarrow\mathcal{T}^{\mathbb{Z}^{2}}$ (“$SR$” for “symbolic representation”). With this map properly defined, we obtain the space of interest, $\mathcal{X}_{\mathcal{P},R}$, as the topological closure of the image of $SR^{\mathbf{v}}$:

Labbé [Lab21] shows that the choice of $\mathbf{v}$ doesn’t actually matter after taking the topological closure, so the space $\mathcal{X}_{\mathcal{P},R}$ is not dependent on $\mathbf{v}$. Figure 3 is worth a thousand words in understanding the essential function of $SR$. In that figure, we see how the points of the orbit of a point $p$ in the torus forms a configuration of Wang tiles.

We pause here to point out that the space $\mathcal{X}_{\mathcal{P},R}$ is fully defined in more technical terms in [Lab21] that facilitate the use of the tools of dynamical systems to prove that if the partition $\mathcal{P}$ and the action $R$ satisfy reasonable technical properties, the partitioned system $(\mathbf{T},\mathbb{Z}^{2},R)$ gives rise to $(\mathcal{X}_{\mathcal{P},R},\mathbb{Z}^{2},\sigma)$ along with morphims going both directions. With such formalisms in place, important properties of the space $(\mathcal{X}_{\mathcal{P},R},\mathbb{Z}^{2},\sigma)$ can be determined. For example, we are interested in knowing when $(\mathcal{X}_{\mathcal{P},R},\mathbb{Z}^{2},\sigma)$ is shift of finite type, which is an important property that a Wang shift has. Another important proptery is minimality of the subshift, which can be deduced from properties of the partition and action.

At this point, it may be believable that $\mathcal{X}_{\mathcal{P},R}$ is a reasonable subset of $\mathcal{T}^{\mathbb{Z}^{2}}$, so configurations of $\mathcal{X}_{\mathcal{P},R}$ can be seen as potential valid configuations (i.e. tilings), but it should not be clear at all how the partition “knows” what the edge matching rules are and how orbits of points correspond to valid tilings that respect the matching rules of the prototiles of $\mathcal{T}$. This is where some magic occurs in the Labbé construction. The magic is that the partition $\mathcal{P}$ needs to be a refinement of four other partitions $\mathcal{P}_{r}$, $\mathcal{P}_{t}$, $\mathcal{P}_{\ell}$, and $\mathcal{P}_{b}$ called side partitions that encode the matching rules of prototiles in $\mathcal{T}$. With these side partitions suitably defined, the main partition $\mathcal{P}$ is defined as $\mathcal{P}=\mathcal{P}_{r}\cap\mathcal{P}_{t}\cap\mathcal{P}_{\ell}\cap\mathcal{P}_{b}$. With $\mathcal{P}$ so defined, and checking that $\mathcal{P}$ has certain necessary dynamical properties, it is argued in [Lab21] that $\mathcal{X}_{\mathcal{P},R}\subseteq\Omega_{\mathcal{T}}$. That is, the configurations generated from orbits of points in the partitioned set $\mathbf{T}$ are valid tilings! We use the word “magic” here to describe the process of finding $\mathcal{P}$ because there is no simple formula for finding such a partition. For a given Wang protoset $\mathcal{T}$, the partition may be found through some educated guesswork and the help of computers. In the next section, we will describe an experimental approach to finding partitions.

Following the method suggested in [GS87, Section 11.1], a tiling by the Penrose rhombs (P3) can be coverted to a tiling by Wang tiles using the patches of tiles shown in Figure 5. This protoset of 24 Wang tiles, as was proven in [Jan21], produces tilings in exact correspondence with the tilings produced by the P3 protoset.

Let the protoseet of Figure 6 be denoted by $\mathcal{T}_{24}$. We may use the computer to produce a large patch of tiling, and then using a matrix

we can produce a dot pattern that reveals a partition for $\Omega_{\mathcal{T}_{24}}$.

From this dot pattern, we can guess at the exact slopes of the lines and produce an exact partition of the torus $\mathbf{T}=\mathbb{R}^{2}\pmod{\mathbb{Z}^{2}}$, viewed as a quotient of the unit square. We have created such a partition in Figure 8. To simplify things, at right Figure 8 we have scaled the partition by a factor of $\varphi$ to obtain a partition $\mathcal{P}_{24}$ , and we may consider the torus $\mathbf{T}=\mathbb{R}^{2}\pmod{\Gamma_{24}}$ where $\Gamma_{24}=\left<(\varphi,0),(0,\varphi)\right>$ with $\mathbb{Z}^{2}$ action $R_{24}$ defined on $\mathbf{T}$ by $R_{24}^{\mathbf{n}}(p)=p+\mathbf{n}\pmod{\mathbf{T}}$. From these ingredients, we can now consider the space $\mathcal{X}_{\mathcal{P}_{24},R_{24}}$ and testable hypotheses concerning that space that lends itself to the techniques developed by Labbé in [Lab21], such as:

• •

  Are configurations in $\mathcal{X}_{\mathcal{P}_{24},R_{24}}$ always valid tilings in $\Omega_{\mathcal{T}_{24}}$? I.e., Is $\mathcal{X}_{\mathcal{P}_{24},R_{24}}\subseteq\Omega_{\mathcal{T}_{24}}$?

• •

  Is $\mathcal{X}_{\mathcal{P}_{24},R_{24}}$ a shift of finite type?

• •

  Is $\mathcal{X}_{\mathcal{P}_{24},R_{24}}$ minimal in $\Omega_{\mathcal{T}_{24}}$?

• •

  Is it true that $\mathcal{X}_{\mathcal{P}_{24},R_{24}}=\Omega_{\mathcal{T}_{24}}$, so that the partition $\mathcal{P}_{24}$ produces all possible Penrose tilings?

• •

  Will $\mathcal{X}_{\mathcal{P}_{24},R_{24}}$ exhibit nonexpansive directions, as corresponding space $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ did for the Jeandel-Rao shift?

We will pursue questions such as these in Appendix B, but for now, we will continue with this experimental approach to finding a few other potential interesting partitions for Wang tile protosets.

The Ammann A2 order-2 aperiodic protoset is shown in Figure 9. Notice at bottom in Figure 9 that the tiles are adorned with red and blue lines called Ammann bars. The matching rule is that when two copies of tiles from A2 meet, Ammann bars of the same color must continue in a straight line. In Figure 10, we see how the matching rules are enforced in a small patch of tiling for the A2 protoset.

The key to converting the A2 protoset to a Wang tile protoset of order 16 is to consider all of the possible rhombi formed from the red lines, with the blue lines serving as the matching rules for the red rhombi, as explained in In [GS87, Section 11.1]. The protoset so produced, $\mathcal{T}_{16}$, is shown in Figure 11.

Again, as we did with the $\mathcal{T}_{24}$, we can use the computer to produce a large patch of tiling, and with the appropriate choice of matrix, we can pull the tiling back onto a torus and see a dot pattern that indicates that a partition underlies the protoset. Indeed, using

we obtain the nice dot pattern shown in Figure 12. Interestingly, we see not only that $A_{24}=A_{16}$, but also that the dot patterns for $\mathcal{T}_{24}$ and $\mathcal{T}_{16}$ align - the pattern for P3 seems to be a refinement of the one for A2 (Figure 13). This is not coincidental. In [GS87, Section 11.1] it is pointed out that $\mathcal{T}_{16}$ can be derived from the Penrose darts and kites (P2) protoset using the idea of Ammann bars.

From the dot pattern obtained for $\mathcal{T}_{16}$, we can determine the exact equations of the lines to get a partition $\mathcal{P}_{16}$, as shown in Figure 14. We have scaled the partitioned torus for $\mathcal{T}_{16}$ by a factor of $\varphi$ (as we did for $\mathcal{T}_{24}$). Define the torus $\mathbf{T}$ identifying opposites sides of this square $[0,\varphi]\times[0,\varphi]$ and define the $\mathbb{Z}^{2}$ action $R_{16}$ on $\mathbf{T}$ by $R_{16}^{\mathbf{n}}(p)=p+\mathbf{n}\pmod{\mathbf{T}}$. We may now consider the space $\mathcal{X}_{\mathcal{P}_{16},R_{16}}$ and consider questions like those outlined in Subsection 4.1.

Before considering more technical theoretical questions about $\mathcal{X}_{\mathcal{P}_{24},R_{24}}$ and $\mathcal{X}_{\mathcal{P}_{16},R_{16}}$, we will test, experimentally, if the configurations in these spaces seem to be valid tilings. In doing so, we will consider orbits of points on the torus that intersect the boundary and points whose orbits do not.

Let $\mathbf{T}=[0,\varphi]\times[0,\varphi]$ where $\mathbf{T}$ has partition $\mathcal{P}_{24}$ and let $p=(0.1,0.2)\in\mathbf{T}$. We then consider the orbit $\mathcal{O}_{R_{24}}(p)\subset\mathbf{T}$. We wish to see if the tiling (or at least a finite portion we can generate) spelled out by $\mathcal{O}_{R_{24}(p)}$ seems to be valid. This process could be automated, but we will do it “by hand”, drawing the orbit in Figure 15 and the corresponding partial configuration in Figure 16. Notice that the partial tiling so generated is valid. Also note that we could easily convert the Wang tiling of Figure 16 into a tiling by the Penrose rhombs (P3) simply by substituting the patches described in Figure 5 for the corresponding Wang tiles. Similarly, in Figure 17 we see part of the orbit $\mathcal{O}_{R_{16}}(p)$ of the point $p=(0.1,0,1)$ in the torus $\mathbf{T}$ partitioned by $\mathcal{P}_{16}$ the corresponding partial tiling by the order-16 Ammann Wang tiles ($\mathcal{T}_{16}$) in 18, which again appears to be valid.

It is not hard to check that the orbit of any point $p\in\mathbf{T}$, under either action $R_{24}$ or $R_{16}$, will be dense in $\mathcal{T}$. Thus, one immediate consequence of $\mathcal{X}_{\mathcal{P}_{24},R_{24}}$ and $\mathcal{X}_{\mathcal{P}_{16},R_{16}}$ being subspaces of $\Omega_{24}$ and $\Omega_{16}$ (respectively) is that the areas of atoms in $\mathcal{P}_{24}$ and $\mathcal{P}_{16}$ correspond to the proportion of tiles of specific kinds in tilings. For example, the area of the atom labeled $0$ in $\mathcal{P}_{16}$ (Figure 14) is $(\varphi-1)^{2}=2-\varphi$, which comprises $(2-\varphi)/\varphi^{2}=5-3\varphi\approx 14.59\%$ of the area of the torus $\mathbf{T}$. This means that in any tiling in $\mathcal{X}_{\mathcal{P}_{16},R_{16}}$, about 14.59% of the tiles will be copies of the prototile $T_{0}$.

Let $\Delta_{16}$ be the union of the boundary lines of the atoms in $\mathcal{P}_{16}$. In this subsection we consider an example where the orbit of a point in $\mathcal{T}$ intersects $\Delta_{16}$, and we note that the details are similar in $\mathcal{P}_{24}$. We will illustrate by example how the ambiguity that arises in assigning tiles to points in the orbit on $\Delta_{16}$ is resolved and we will see a certain interesting phenomenon in the dynamics of $\mathcal{X}_{\mathcal{P},R}$.

Let $p=\mathbf{0}=(0,0)\in\mathbf{T}$ and let $q=(1/2,(3/2)\varphi-1)\in\mathbf{T}$. Notice that $p$ lies at the intersection of lines of all 4 slopes of boundary lines in $\mathcal{P}$ since this torus $\mathbf{T}$ is formed as a quotient by identifying opposite sides. Also notice that $q$ lies on a boundary segment of slope $\varphi$ connecting $(0,\varphi-1)$ and $(\varphi-1,1)$. Consider the orbit $\mathcal{O}_{R_{16}}(p)$. Using the computer, inside of a finite window of size $[-40,40]\times[-40,40]$, we can check which points $\mathbf{n}\in\mathbb{Z}^{2}$ satisfy $R_{16}^{\mathbf{n}}(\mathbf{0})\pmod{\mathbf{T}}\in\Delta_{16}$, and a plot of these points $\mathbf{n}$ is shown in Figure 19(a). Similarly, inside the same size window, we plot the points $\mathbf{n}\in\mathbb{Z}^{2}$ satisfying $R_{16}^{\mathbf{n}}(q)\pmod{\mathbf{T}}\in\Delta_{16}$ in Figure 19(b). Interestingly, we find that the points $\mathbf{n}\in\mathbb{Z}^{2}$ where the orbit falls on the boundary forms straight strips of points, sometimes a single strip (when the orbit touches lines in the boundary of just one slope) or four separate strips (when the orbit touches lines in the boundary of different slopes). The directions of the strips are called the nonexpansive directions of $\mathcal{X}_{\mathcal{P}_{16},R_{16}}$. In [LMM22], the authors provide methodology for determining the exact directions of such nonexpansive directions, and we leave that topic for the reader to explore further. We only mention here that these nonexpansive directions correspond to what are commonly called Conway worms in Penrose tilings [Gar97]; nonexpansive directions provides a very nice and more general explanation for such behavior in some tilings.

In either situation, the ambiguity in the corresponding tiling can be resolved by choosing a direction to determine what tile to place at the points $\mathbf{n}$ such that $R^{\mathbf{n}}_{16}(p)$ lies on $\Delta_{16}$. For example, if we again consider the orbit $\mathcal{O}_{R_{16}}(q)$ where $q$ is as in Figure 19(b), let us specify a direction vector $\mathbf{v}=(-1,1)$. Notice that no line segment in $\Delta_{16}$ is parallel to $\mathbf{v}$. Now, at each point of $\mathbf{n}$ where $\mathcal{O}_{R_{16}}(q)$ intersects $\Delta_{16}$, place vector $\mathbf{v}$ with its initial point at $\mathbf{v}$. Because $\mathbf{v}$ is not parallel to any of the segments comprising $\Delta_{16}$, then $\mathbf{v}$ is pointing from $\mathbf{n}$ into one of the atoms $P_{i}\in\mathcal{P}_{16}$. Thus, we can place a copy of prototile $T_{i}$ at $\mathbf{n}$. The other choice is use $-\mathbf{v}$ to determine the choice of tile at each such point $\mathbf{n}$, which points toward the region on the opposite side of the boundary segment as does $\mathbf{v}$. This is illustrated in Figures 20 and 21.