1 Introduction

Parallelogram tilings, in particular rhomb tilings, are a fundamental family in the mathematical study of tilings, in both aperiodic order with its links to quasicrystals and in the study of random tilings. In addition, they provide visual models for a wide range of mathematical structures.

The most famous aperiodic parallelogram tiling is certainly the Penrose tiling [13, 15, 23] in the version with rhombic tiles. Many other aperiodic tilings use parallelograms as tiles, including the Ammann–Beenker tilings [2, 5], or more generally, all canonical projection tilings [11, 18]. Further examples arise in the study of geometric representations of substitutive dynamical systems, see for instance [24]. Such models are of particular importance in the study of mathematical quasicrystals [26] and systems with long-range aperiodic order [3, 22].

In random tilings parallelogram tilings are also of importance. Usually finite tilings are considered, lozenge tilings of polygons [7, 21], or parallelotope tilings of zonotopes [8, 9], but infinite random parallelogram tilings are also studied [6]. In this paper we focus on infinite tilings of the plane, but some results are also valid and relevant for finite tilings, for instance Lemmas 2.3 and  2.4.

Parallelogram tilings also turn up in the study of pseudo-line arrangements in combinatorics (for example [1, 16]), as a visualisation of reduced words in Coxeter groups [10] and, through random tilings and dimer models, they link to a wide variety of physical models [28]. The use of parallelogram tilings in a wide variety of mathematics is discussed further on Math Overflow [27].

Interestingly, there are aperiodic tilings of the plane with polygonal tiles where the tiles occur in an infinite number of orientations: The Pinwheel Tiling [25] is an example of a tiling using only one kind of tile (a triangle with edge lengths 1, 2, and \(\sqrt{5}\)) where the tiles occur in infinitely many orientations throughout the tiling. A comprehensive study of this phenomenon is [12]. There are also tilings using two kinds of quadrilaterals in which the tiles occur in infinitely many orientations [4], see for instance the Kite Domino Tiling in the online resource [13]. (Click “Kite-Domino” on the main page for information about the Kite-Domino Tiling, and then “Infinite rotations” for a list of other examples.) Our main result states that such a behaviour is impossible if all tiles are parallelograms.

By definition, a parallelogram has two pairs of parallel edges. In a parallelogram tiling therefore there are natural lines of tiles linked each sharing a parallel edge with the next. These lines are called worms and were used by Conway in studying the Penrose tilings [17]. We will follow Conway to call these lines worms. Figure 3 shows three (finite pieces of) such worms. In this paper, we consider the structure of these worms and use them to prove results on parallelogram tilings. In particular, our main result states that no parallelogram tiling using finitely many different parallelograms can have tiles in infinitely many orientations.

This paper is organised as follows: the first section defines the necessary terms and notations. The next section is dedicated to the proof of our main result Theorem 2.5. It contains several results which may be of interest on their own in the study of parallelogram tilings, for instance the Crossing Lemma 2.3 or the Travel Lemma 2.4. The last section contains some additional remarks.

A tiling \(\mathcal{T }\) of the plane \(\mathbb{R }^2\) is a packing of \(\mathbb{R }^2\) which is also a covering of \(\mathbb{R }^2\). Usually, the tiles are nice compact sets, and the tiling is a countable collection of tiles \(T_i: \mathcal{T }= \{ T_1, T_2, \ldots \}\). The covering condition can then be formulated as \(\bigcup _i T_i = \mathbb{R }^2\), and the packing condition as \(\mathrm{int}(T_i) \cap \mathrm{int}(T_j) = \varnothing \) for \(i \ne j\), where \(\mathrm{int}(A)\) denotes the interior of \(A\).

The set of congruence classes of tiles in a tiling \(\mathcal{T }\) is called protoset of \(\mathcal{T }\). Its elements are called prototiles of \(\mathcal{T }\). A tiling is called locally finite, if each ball \(B \subset \mathbb{R }^2\) intersects only a finite number of tiles of \(\mathcal{T }\). A tiling \(\mathcal{T }\) is called vertex-to-vertex, if all prototiles are polygons, and for all tiles in \(\mathcal{T }\) holds: If \(x \in \mathbb{R }^2\) is vertex of some tile \(T \in \mathcal{T }\), then \(x \in T^{\prime } \in \mathcal{T }\) implies \(x\) is a vertex of \(T^{\prime }\).

General Assumption Throughout the paper, only locally finite vertex-to-vertex tilings are considered. Moreover, the the prototiles will always be parallelograms. In particular, all tiles have nonempty interior.

Definition 1.1

Let \(T\) be a tile in a parallelogram tiling, and let \(e\) be one of its edges. There is a unique tile \(T_1\) sharing the edge \(e\) with \(T\). Let \(e_1 \ne e\) denote the edge of \(T_1\) parallel to \(e\). There is a unique tile \(T_2\) sharing this edge with \(T_1\). Repeating this yields a (one-sided) infinite sequence of tiles \(T, T_1, T_2, \ldots \) uniquely given by \(T\) and \(e\). Repeating this on the edge of \(T\) which is opposite of \(e\) yields a further (one-sided) infinite sequence of tiles \(T_{-1}, T_{-2}, \ldots \). In this way, \(T\) and \(e\) define a unique biinfinite sequence \(W:=\ldots T_{-2}, T_{-1}, T, T_1, T_2, \ldots \). We will call this sequence the worm given by T and e, and any such sequence just a worm. The intersection of two worms \(W, W^{\prime }\) is defined as \(W \cap W^{\prime } = \{ T \, | \, T \in W, T \in W^{\prime } \}\).

Worms are also called de Bruijn lines [9] or ribbons [7]. Worms correspond to pseudo-line configurations [1, 16], and to grid lines used in the construction of canonical projection tilings by the multigrid method [14].

Definition 1.2

Let \(e\) be some edge in a parallelogram tiling. The set of all worms which are defined by edges parallel to \(e\) is called the worm family (given by \(e\)).

2 Proof of the Main Result

In this section we collect some basic results on worms, which will be used in the next section. The first result states that a worm cannot bend “too much”. In order to state this precisely, we define the open cone \(C(x,\alpha )\) with axis \(x \in \mathbb{R }^2\) and angle \(\alpha \in [0,\pi [\) by \(C(x,\alpha ) := \{ z \in \mathbb{R }^2 \, | \, |\angle (x,z)| < \alpha \}\), where \(\angle (x,z) = \arccos \frac{ \langle x,z \rangle }{\Vert x\Vert \Vert z\Vert }\) denotes the angle between \(x\) and \(z\).

Lemma 2.1

(Cone Lemma) Let \(\mathcal{T }\) be some parallelogram tiling, with finite or infinite protoset. Denote the infimum of all interior angles of the prototiles by \(\alpha \). Let \(W\) be a worm given by some tile \(T\) and some edge \(e\). Denote the two endpoints of \(e\) by \(x_1\) and \(x_2\). If \(\alpha >0\) then \(W\) is contained in

$$\begin{aligned} \mathbb{R }^2 \backslash \left(C_1 \cup C_2 \right), where \ C_1:= x_1+C(x_1-x_2, \alpha ), \, C_2 := x_2+C(x_2-x_1, \alpha ). \end{aligned}$$

If \(\alpha =0\) then \(W\) is contained in \(e \cup \big ( \mathbb{R }^2 \setminus \mathrm{aff}(e) \big )\), where \(\mathrm{aff}(e)\) denotes the affine span of \(e\).

Proof

By inspection of Fig. 1 we observe that the worm defined by the edge \(e\) cannot intersect the cones (shaded) \(C_1\) and \(C_2\), where \(\alpha >0\) is the smallest interior angle of the prototiles. Using the prototile with the smallest angle \(\alpha \) alone, the worm can line up along the boundary line of one of the cones at best.

Fig. 1
figure 1

The worm defined by the edge \(e\) cannot intersect the open cones \(C_1\) or \(C_2\)

Full size image

If \(\alpha =0\), the open cones are empty. Nevertheless, since any tile has nonempty interior, the particular tile in the worm \(W\) containing \(e\) as an edge has some positive interior angle. Then, by the same reasoning as above, \(W\) cannot touch the line \(\mathrm{aff}(e)\) apart from \(e\). \(\square \)

The last lemma shows also that a worm can have no loop. We state this result as a an additional lemma.

Lemma 2.2

(Loop Lemma) A worm has no loop.

Lemma 2.3

(Crossing Lemma) Two worms cross at most once. Worms in the same family never cross.

In other words, the intersection of any two worms is empty or consists of a single tile. The intersection of two different worms defined by parallel edges is empty.

Proof

Consider two worms in distinct families, and put their defining edges together at a point (see Fig. 2). Since the tiles are parallelograms, and the interior angles of parallelograms are less than \(\pi \), the two worms can only cross in the direction where the edges form an angle less than \(\pi \). After crossing this direction is reversed. Thus no further crossing is possible.

Fig. 2
figure 2

Two worms can cross at most once

Full size image

If the two worms are in the same family the crossing tile would have to have all four edges parallel, but no tile (with positive area) can have this property. \(\square \)

Lemma 2.4

(Travel Lemma) Let \(\mathcal{T }\) be some parallelogram tiling, with finite or infinite protoset. Denote the infimum of all interior angles of the prototiles by \(\alpha \). If \(\alpha > 0\), then any two tiles \(S,T\) in \(\mathcal{T }\) can be connected by a finite sequence of tiles which are contained in a finite number of worms. Moreover, less than \(\lceil 2 \pi / \alpha \rceil \) different worms are needed. (Here \(\lceil x \rceil \) denotes the smallest \(k \in \mathbb{Z }\) such that \(k \ge x\).)

In plain words: If the prototiles are not arbitrarily thin, then we can always travel from tile \(S\) to tile \(T\) by walking on worms, with finitely “turns”; that is, with finitely many changes from one worm to another one. The proof stresses the fact that tiles cannot be arbitrarily thin.

Proof

We start from tile \(S\), and choose \(W_0\), one of the two worms passing through \(S\). As, by Lemma 2.2 \(W_0\) is not a loop, an infinite number of worms cross \(W_0\). The tile \(T\) will either lie on one of these worms or between two of them. (The latter situation is indicated in Fig. 3.)

Fig. 3
figure 3

Starting from a tile \(S\) and worm \(W_0\), there are a set of worms crossing \(W_0\). A tile \(T\) that does not lie on one of these worms must lie between two. We label the worm further from \(S\) with \(W_1\)

Full size image

If \(T\) lies on one of the worms we are done. Otherwise there are two worms, defined by adjacent tiles in \(W_0\), such that \(T\) lies between these two worms. Let \(W_1\) be the worm further from \(S\) and \(S_1\) be the tile given by the intersection of \(W_0\) and \(W_1\). Again \(T\) either lies on a worm crossing \(W_1\) or between two worms. If \(T\) does lie on such a worm we are done. Otherwise call the worm further from \(S_1\) \(W_2\).

Using this method we obtain a sequence of worms \(W_0, W_1, \ldots \). Note that we cannot visit infinitely many tiles of the same worm: in this case we would have found infinitely many worms passing between \(S\) and \(T\), which is impossible, since worms cannot be arbitrarily thin.

Thus, we obtain a sequence of finite pieces of worms by our method. We proceed to show that we have to arrive at \(T\) in this way after finitely many turns.

By assumption we have \(\alpha > 0\). Thus, after at most \(k:=\lceil 2 \pi / \alpha \rceil \) turns we completed a full turn by \(2 \pi \) or more. Note that, by construction, the tile \(T\) is always on the right hand side of the current worm, or always on the left hand side. Without loss of generality, let \(T\) be always on the right hand side of the current worm. Now, let us travel further on the current worm \(W_k\), say. Either \(S\) is contained in \(W_k\) (Case 1), or on the left hand side of it (Case 2), or on the right hand side of it (Case 3). In the first two cases, we have completed a loop (Case 2 is indicated in Fig. 4).

Fig. 4
figure 4

Case 2: the constructed worm sequence encircles \(T\)

Full size image

Assume we have completed a loop. By construction, \(T\) lies inside this loop. Since, by construction, our sequence of worms is not crossed by any worm defined by \(T\), the two worms defined by \(T\) must be contained entirely inside this loop. Since worms are infinite, and tiles are not arbitrarily small, this is impossible.

It remains to consider Case 3: \(S\) is on the right hand side of our current worm \(W_k\). There are two subcases: Either \(W_k\) reaches the first worm \(W_0\), defined by \(S\). In this case we have closed a loop again, yielding a contradiction. Or \(W_k\) avoids the first worm \(W_0\). Now we utilize the Cone Lemma 2.1 to show that this case is impossible, too. Let us denote the tile where we entered \(W_k\) by \(S_k\). By Lemma 2.1, the cone \(C_0\) pointing outward our sequence (to the left) defined by \(S\) and the edge defining \(W_0\) cannot contain tiles of \(W_0\). Similarly, the cone pointing outward defined by \(S_k\) and the edge defining \(W_k\) cannot contain tiles of \(W_k\).

Either \(C_k\) intersects \(W_0\), then \(W_k\) intersects our worm sequence, closing a loop. Otherwise, since \(k \ge 2\pi /\alpha \), the axis of \(C_k\) is tilted towards \(S\), thus \(C_0\) and \(C_k\) have non-empty intersection. (This situation is indicated in Fig. 5.) Neither \(W_0\) nor \(W_k\) can intersect \(C_0 \cap C_k\). \(W_0\) cannot intersect \(C_0\), and \(W_k\) cannot intersect \(C_k\), thus \(W_0\) and \(W_k\) have to intersect each other. This closes a loop, which yields a contradiction.

Fig. 5
figure 5

Case 3: the constructed worm sequence does not entirely encircle \(T\), but nevertheless \(W_0\) and \(W_k\) have to cross (since the cones \(C_0\) and \(C_k\) are tilted towards each other, \(W_0\) cannot intersect \(C_0\), and \(W_k\) cannot intersect \(C_k\))

Full size image

Thus our constructed worm sequence cannot avoid \(T\). Thus, \(T\) is reached from \(S\) by using (finitely many) tiles contained in finitely many (\(k\) or less) worms. \(\square \)

Theorem 2.5

Let \(\mathcal{T }\) be a parallelogram tiling with a finite protoset. Then each prototile occurs in a finite number of orientations in \(\mathcal{T }\). Moreover, the number of orientations of each prototile is bounded by a common constant.

Proof

This is now an immediate consequence of Lemma 2.4: Let \(\alpha \) denote the minimal interior angle in the prototiles. Fix some tile \(S\) in the tiling. Any other tile \(T\) of the same shape can be reached from \(S\) by a sequence of worms with at most \(k = \lceil 2 \pi / \alpha \rceil \) turns. Let there be \(m\) prototiles altogether. If we can connect \(S\) with \(T\) without any turn at all (i.e., \(S\) and \(T\) are contained in the same worm) there are \(2\) possibilities how \(T\) is oriented with respect to \(S\). (\(T\) can be a translated copy of \(S\) or a reflected copy of \(S\), yielding two different orientations unless \(S\) and \(T\) are squares.) If we can connect \(S\) with \(T\) using one turn, there are \(2m\) possibilities how \(T\) is oriented with respect to \(S\): \(m\) possibilities arising from the choice of the prototile defining the turn, and two possibilities for any such choice (left- or right-handed). Analoguously, for any turn there are \(2m\) possibilities of changing the orientation. Thus, there are \((2m)^{\ell }\) possibilities if we need exactly \(\ell \) turns. Altogether, this leaves

$$\begin{aligned} \begin{aligned} N&:= 2 + 2m + (2 m)^2 + \cdots + (2 m)^k \\&= 1+ \sum _{n=0}^k (2m)^k = \frac{1}{2m-1} \big ( (2m)^{k+1} +2m -2 \big ) \end{aligned} \end{aligned}$$

as an upper bound for the number of different orientations of \(T\) with respect to \(S\). This is true for any tile \(T\) congruent to \(S\). Thus there are at most \(N^2\) possibilities for the orientation of any two congruent tiles \(T, T^{\prime }\) to each other. \(\square \)

At this point one may ask how the result generalises to prototiles being centrally symmetric convex polygons.

Corollary 2.6

Let \(\mathcal{T }\) be a tiling with a finite protoset, and let all prototiles be centrally symmetric convex polygons. Then each prototile occurs in a finite number of orientations in \(\mathcal{T }\).

Proof

It is known that each centrally symmetric convex polygon can be tiled by parallelograms, see [19], Theorem 1. Any tiling \(\mathcal{T }\) by centrally symmetric convex polygons with tiles in infinitely many orientations could be refined into a tiling by parallelograms in infinitely many orientations, in contradiction to Theorem 2.5. \(\square \)

3 Remarks

It is easy to see that Theorem 2.5 becomes false if we allow an infinite number of prototiles, even if the minimal interior angle \(\alpha \) is bounded away from 0. Nevertheless, the Travel Lemma still holds in this case: any two tiles can be connected by a (finite) sequence of tiles which is contained in a finite number of worms.

The upper bound obtained in the proof of Theorem 2.5 is not optimal. For instance, the angle changes induced by turns are not always independent: the pair of turns induced by a tile of type 1 (right-handed, say) followed by a turn induced by a tile of type 2 (right-handed) yields the same relative orientation as the pair of turns induced by a tile of type 2 (right-handed) followed by a turn induced by a tile of type 1 (right-handed). However, we do not see how to bring this upper bound below \(O\big ( (2m)^k \big )\).

Throughout the paper we made the general assumption that all tilings are vertex-to-vertex, and that all tilings are locally finite. The requirement of local finiteness is essential, otherwise the Travel Lemma would become false: we can no longer guarantee that any two tiles can be connected by a finite sequence of tiles at all.

The requirement that the tilings are vertex-to-vertex allows for the definition of worms. However, if the tilings are locally finite, but not vertex-to-vertex, then we can dissect each parallelogram into finitely many parallelograms such that the new tiling is vertex-to-vertex. Thus, all results apply also to locally finite parallelogram tilings which are not vertex-to-vertex.

One may ask whether Theorem 2.5 can be generalized to arbitrary centrally symmetric polygons. By the results in [19] and [20], a centrally symmetric polygon can be tiled by parallelograms if and only if it satisfies certain technical conditions: the “balance condition” and the “convex crossing condition”, see Theorem 2 in [19]. Thus Theorem 2.5 generalizes to any tiling by prototiles fulfilling these conditions, along the lines of the proof of Corollary 2.6. A simple example of a centrally symmetric polygon which does not fulfil these conditions—thus can not be tiled by parallelograms—is a bow-tie (or hour-glass) shaped centrally symmetric nonconvex hexagon. For such tiles the methods of this paper break down: the Cone Lemma is not longer true in this case, thus the Loop Lemma cannot be derived from it. Also the Crossing Lemma is false for such tiles: it is easy to construct counterexamples to the Crossing Lemma using the mentioned bow-tie shaped hexagons plus some parallelograms. This leads us to the formulation of the following problem.

Problem Is there a locally finite tiling by centrally symmetric polygons where the tiles occur in an infinite number of orientations?

Equivalent results to those proved in this paper should apply to tilings of \(\mathbb{R }^d\,(d \ge 2)\) by parallelotopes. In order to generalise Lemma 2.4, however, we need to enclose some tile \(T\) with a finite collection of tiles. This would require further objects. In addition to the pseudo-lines for worms, one must also include pseudo-planes (and pseudo-hyperplanes) made up of linked parallelotopes sharing a parallel edge in common.