Abstract
Inspired by the modelization of 2D materials systems, we characterize arrangements of identical nonflat squares in 3D. We prove that the fine geometry of such arrangements is completely characterized in terms of patterns of mutual orientations of the squares and that these patterns are periodic and onedimensional. In contrast to the flat case, the nonflatness of the tiles gives rise to nontrivial geometries, with configurations bending, wrinkling, or even rolling up in one direction.
1 Introduction
The serendipitous isolation of graphene in 2004 [25] attracted enormous interest on the physics of 2D materials systems. Driven by their fascinating electronic and mechanical properties [34], research on 2D systems is currently witnessing an exponential growth. Beyond graphene [2, 16], 2D material systems are continuously synthetized and investigated [7, 9, 19, 36] and findings are emerging at an always increasing pace, ranging from fundamental understanding to applications [1].
Free standing 2D material samples are often not flat, but rather present rippling patterns at specific length scales [18]. The origin of such nonflatness is currently debated, one possible explanation being the instability of perfectly flat arrangements at finite temperatures, as predicted by the classical MerminWagner theory [22, 23]. In the case of graphene, ripples have been experimentally observed [20, 24], computationally investigated [12], and analytically assessed [13, 14]. The phenomenon is however not restricted to graphene, and surface rippling has been detected in other 2D systems as well [5, 30]. Understanding the global geometry of 2D materials is of the greatest importance, as flatness is known to influence crucially the electronic, thermal, and mechanical behavior of these systems [8, 10, 33, 35].
In this paper, we tackle the question of flatness of 2D systems with square symmetry. Our interest is theoretical and our arguments are not tailored to a specific material system. Still, we remark that squarelike 2D crystals have been predicted in selenene and tellurene [32]. We formulate the problem in the setting of molecular mechanics [3, 17, 26] by associating to each point configuration a scalar configurational energy and focusing on its ground states in the quest for optimal geometries [4, 15]. In the squaresymmetric case, each atom has four first neighbors and the topology of the configuration is that of the square lattice \({\mathbb {Z}}^2\) [21]. The configurational energy is assumed to feature both two and threebody effects [6, 27, 29], depending on bond lengths (distances between atoms) and angles between bonds, respectively. We present conditions ensuring that global minimizers of the configurational energy have all bonds of equal length, all angles formed by bonds to first neighbors of equal amplitude \(\theta ^*\), and the four first neighbors of each atom are coplanar. As a result, minimal cycles of four atoms form regular squares featuring equal sides and equal angles \(\theta ^*\), see Fig. 1. Such identical squares arrange then in an infinite 3D configuration, which under the above provisions we call admissible and which we interpret as the actual geometry of the crystal.
The goal of this paper is to classify all admissible configurations, namely all possible 3D arrangements of identical regular squares. In case the squares are flat, namely if \(\theta ^*=\pi /2\), the result is straightforward: the only configuration of flat squares where all first neighbors of each atom are coplanar is the plane. In order to tackle genuinely 3D geometries, we hence need to focus on the case \(\theta ^*<\pi /2\) instead, which induces nonflatness, as per Fig. 1.
Our main result is a complete characterization of admissible arrangements of identical regular nonflat squares in 3D, see Theorem 2.8. We prove in particular that admissible configurations can bend, wrinkle, and roll in one direction and that such flexural behavior is completely characterized by specifying a suitably defined section of the configuration in the bending direction, see Fig. 4 below. More precisely, one classifies patches of four squares sharing an atom (4tiles) in six different classes, in terms of their mutual orientation, see Fig. 6. We prove that just three of these classes actually give rise to admissible configurations, that the whole geometry is specified by knowing the pattern of such classes, and that such pattern is periodic.
One can visualize the square in Fig. 1 as (the boundary of) a nonflat tile. Our result can hence be interpreted as a classification of all possible tilings with such nonflat tiles under the condition that the four neighbors of each atom are coplanar. The relevance of this coplanarity condition is revealed by considering the limiting flat case. In case tiles are flat and the four neighbors of each atom are coplanar, the only possible tiling is the plane. By dropping the coplanarity requirement, we however allow for tilings ensuing from foldings of the reference square lattice \({\mathbb {Z}}^2\) along a set of parallel coordinate directions. Thus, the coplanarity requirement serves the purpose of excluding the effect of the symmetry of the reference lattice on the onset of nontrivial geometries.
In the case of hexagonal symmetry, the characterization of global arrangements of regular nonflat hexagons has been obtained in [13, 14]. To some extent, the results in this paper for squares are akin to the hexagonal case, for in both cases the arrangement shows some distinguished onedimensional patterning. Compared with the hexagonal setting, the present squaresymmetric case is however much more involved. This is an effect of the different symmetry of the underlying reference lattices. In the square case, arguments require to consider the detailed geometry of patches of up to sixteen neighboring squares, which makes the combinatorial picture much richer.
The paper is organized as follows. Section 2 is devoted to the statement of our main results. The molecularmechanical model is discussed first and the detailed geometry of ground states is assessed. A first description of admissible configurations is presented in Theorem 2.2. We then introduce the concept of 4tile and of its type, collect all possible types and classes, and discuss the possibility of attaching two 4tiles by analyzing the corresponding boundary, see Lemma 2.6. This eventually paves the way to the statement of our main result, namely the characterization of Theorem 2.8. Section 3 is entirely devoted to the proof of the main result, hinging both on combinatorial and geometrical arguments. Some proofs are postponed to the Appendix in order to enhance the readability of the arguments.
2 The Setting and Main Results
2.1 Ground States of Configurational Energies
We focus on threedimensional deformations \(y:{\mathbb {Z}}^2 \rightarrow {\mathbb {R}}^3\), defined on the twodimensional reference lattice \({\mathbb {Z}}^2\). For any open subset \(\Omega \subset {\mathbb {R}}^2\) we define the configurational energy of a deformation on \(\Omega \) by
where
denotes the set of nearestneighbors and
is the set of closest nexttonearestneighbors. Moreover, by \(\measuredangle y(x)\, y(x') \, y(x'')\) we denote the bond angle in \([0,\pi ]\) at \(y(x')\) formed by the the vectors \(y(x)y(x')\) and \(y(x'') y(x')\), where the set of triplets \(T(\Omega )\) is defined by
The factor 1/2 reflects the fact that bonds \(\lbrace y(x), y(x') \rbrace \), \((x,x') \in N_1(\Omega ) \cup N_2(\Omega )\), and bond angles \(\measuredangle y(x)\, y(x') \, y(x'')\) appear twice in the corresponding sums. Let us point out that in order to take surface effects at \(\partial \Omega \) properly into account, bonds \(\lbrace y(x), y(x') \rbrace \) are only counted once if \(\{ x, x'\} \in N_1(\Omega )\) and either \(x\in \partial \Omega \) or \(x' \in \partial \Omega \), or if \(\{x,x'\} \in N_2(\Omega )\) and \(x \in \partial \Omega \) or \(x' \in \partial \Omega \). Bonds where \(\{x,x'\} \in N_1(\Omega )\) with \(x \in \partial \Omega \) and \(x'\in \partial \Omega \) are not counted at all. This asymmetry of counting bonds is motivated by the specific choice of the cell energy, see Sect. A.5.
We assume the twobody interaction potential \(v_2:{\mathbb {R}}^+\rightarrow [1,\infty )\) to be of shortrange repulsive and longrange attractive type. In particular, we assume that \(v_2\) is continuous and attains its minimum value only at 1 with \(v_2(1) = 1\). Moreover, we suppose that \(v_2\) is decreasing on (0, 1), increasing on \([1,\infty )\), and that \(v_2\) is continuously differentiable on (1, 2] with \(v_2' >0\) on (1, 2]. The threebody interaction density \(v_3:[0,\pi ] \rightarrow [0,\infty )\) is assumed to be strictly convex and smooth, with \({v}_3(\pi ) = 0\).
In the following, we will be interested in minimizing the energy of a configuration on the whole reference lattice. To this end, we define the normalized energy of \(y:{\mathbb {Z}}^2 \rightarrow {\mathbb {R}}^3\) by
where \(Q_m \subset {\mathbb {R}}^2\) is the open square centered at 0 with sidelength 2m. A deformation is called a ground state if it minimizes the energy E.
For a fine characterization of the minimizers, some additional qualification on \(v_2\) and \(v_3\) will be needed. More precisely, we suppose that there exist small parameters \(\eta , \varepsilon >0\) such that
Properties (2.6)–(2.7) entail that firstneighbor bond lengths range between \(1\eta \) and \(1+\eta \), whereas (2.8) ensures that bond angles are not significantly smaller than \(\pi /2\). Eventually, assumptions (2.9)–(2.11) yield that the contributions of first and second neighbors are strong enough to induce local geometric symmetry of ground states, i.e., bonds and bond angles will be constant, see (2.12)–(2.14) below.
Note that the assumptions (2.6)–(2.11) are compatible with a choice of a density \(v_2\) growing sufficiently fast out of its minimum. In particular, the quantitative Lennard–Joneslike case of Theil [28] (see also [11, 31]) can be reconciled with assumptions (2.6)–(2.7), upon suitably choosing densities and parameters. Let us however remark that the specific form of (2.6)–(2.11) is here chosen for the sake of definiteness and simplicity. Indeed, these assumptions could be weakened, at the expense of additional notational intricacies. Under the above assumptions we have the following result, where we set \(N_1 :=N_1({\mathbb {R}}^2)\), \(N_2 :=N_2({\mathbb {R}}^2)\), and \(T :=T({\mathbb {R}}^2)\) (see (2.2)–(2.4)).
Proposition 2.1
(Ground states). Let \(v_2\) and \(v_3\) be the aboveintroduced two and threebodypotentials satisfying assumptions (2.6)–(2.11). For \(\eta \) small enough and \(\varepsilon = \varepsilon (\eta )\) small enough there exist \(\ell \le 1\), \(\theta <\pi /2\), and \(\delta _\theta < \pi \) only depending on \(v_2\) and \(v_3\) such that a deformation \(y:{\mathbb {Z}}^2 \rightarrow {\mathbb {R}}^3\) is a ground state of the energy E if and only if y satisfies
and
as well as
Here, the conditions \((x,x'') \in N_2\) and \((x,x'') \notin N_2\) correspond to the case that the vectors \(xx'\) and \(x''x'\) form an angle \(\pi /2\) or \(\pi \), respectively, in the reference lattice. We will see later that \(\delta _\theta \) is uniquely determined by \(\theta \) due to a geometric compatibility condition, see Lemma 2.4 below.
The proof of Proposition 2.1 is similar to the one in [13, Proposition 3.1] and is postponed to Appendix A.5. At this stage, let us just comment on the effect of condition \(v_2' > 0\) in a neighborhood of \(\sqrt{2}\), see (2.11), which guarantees that \(\theta \) is strictly smaller than \(\pi /2\). Indeed if \(v_2' = 0\) in a neighborhood of \(\sqrt{2}\), we would obtain \(\ell = 1\) and \(\theta = \pi /2\), i.e., \(y({\mathbb {Z}}^2)\) would coincide with \({\mathbb {Z}}^2\) up to isometries. For \(\theta <\pi /2\) instead, ground states exhibit interesting nontrivial geometries. The aim of this paper is precisely that of characterizing these nontrivial geometries.
2.2 Necessary Conditions for Admissibility
Deformations \(y:{\mathbb {Z}}^2 \rightarrow {\mathbb {R}}^3\) satisfying the conditions (2.12)–(2.14) are called admissible. Without restriction, we suppose for notational convenience that \(\ell =1\). Indeed, this can be achieved by replacing y by \(\frac{1}{\ell }y\) without effecting the geometry of admissible configurations.
Obviously, conditions (2.12)–(2.13) constrain the local geometry of configurations: let \(\{x_1, x_2, x_3, x_4\}\) be a simple cycle in \({\mathbb {Z}}^2\), called a reference cell, where here and in the following the labeling is counterclockwise and counted modulo 4. The image via y is the simple cycle \(\{y_1, y_2, y_3, y_4\}\), where \(y_i = y(x_i)\), called an optimal cell. Since \(\theta < \pi /2\) from (2.13), optimal cells are not flat. Indeed, the sum of interior angles is strictly less then \(2\pi \), i.e., \(\sum _{i = 1}^{4} \measuredangle y_{i1} \, y_i \, y_{i+1} = 4 \theta < 2 \pi \), see also Fig. 2.
The kink of an optimal cell can equivalently be visualized as occurring along the diagonal \(x_3  x_1\) or along the diagonal \(x_4  x_2\) of the corresponding reference cell. We set \(m_1 := (y_1 + y_3)/2\) and \(m_2 := (y_2 + y_4)/2\) and define \(p := m_1  m_2\). Let n be the normal vector of the triangle formed by \( y_1\), \(y_2\), and \(y_4\), in direction \((y_2  y_1) \times (y_4y_2)\). Then, we say that the optimal cell is of form \(\diagdown \) if \(p \cdot n > 0\) and of form \(\diagup \) if \(p \cdot n < 0\), see Fig. 2. An optimal cell of any form can be transformed into a cell of the other form simply via a rotation by \(\pi /2\) along the vector p or via a reflection with respect to the plane with normal p.
Our goal is to provide a complete characterization of admissible configurations. In a first step, we will present necessary conditions for admissibility in terms of optimal cells. To obtain a complete characterization, we will subsequently present a refined formulation in terms of socalled 4tiles, namely, \(2\times 2\) groups of optimal cells, see Sect. 2.5. To state our first main result, we need to introduce some further notation.
Form function. Given a reference cell \(\{x_1, x_2, x_3, x_4\}\) labeled in such a way that for the lowerleft corner \(x_1\) we have \(x_1 = (s,t)\), we define the barycenter z of the reference cell via \(z(s,t) := (1/2 + s, 1/2 + t)\). Thus, \(z({\mathbb {Z}}^2)= {{\mathbb {Z}}^2}^*\), where \({{\mathbb {Z}}^{2}}^*\) denotes the dual lattice of \({\mathbb {Z}}^2\). For an admissible configuration y, we define the form function on the dual lattice \(\tau _y :{{\mathbb {Z}}^{2}}^* \rightarrow \{\diagdown , \diagup \}\) as the map assigning to each reference cell the form of the optimal cell in the deformed configuration. In other words, the deformation y maps a reference cell with barycenter z(s, t) to an optimal cell of form \(\tau _y(z(s,t))\). In the sequel, we simply write \(\tau \) for notational convenience.
Incidence angles. We define the diagonals \(d_1 = (1,1)\) and \(d_2 = (1,1)\). For \(i=1,2\), we indicate signed incidence angles along the diagonal \(d_i\) for each bond of the configuration via the mappings \(\gamma _i:(({\mathbb {Z}}+1/2) \times {\mathbb {Z}}) \cup ({\mathbb {Z}}\times ({\mathbb {Z}}+1/2)) \rightarrow [\pi ,\pi ]\) defined as follows: first, for \(s, t \in {\mathbb {Z}}\), \((s+1/2,t)\) parametrizes the horizontal bond in the reference lattice connecting (s, t) and \((s+1,t)\), and \((s,t+1/2)\) parametrizes the vertical bond in the reference lattice connecting (s, t) and \((s,t+1)\), see Fig. 3A. In the following, we explicitly give the definition of the incidence angle \(\gamma _i(s+1/2,t)\), \(i=1,2\), for horizontal bonds. The definition associated to vertical bonds follows analogously, up to a rotation of the reference lattice by \(\pi /2\).
Consider a horizontal bond parametrized by \((s+1/2,t)\), which is shared by the two cells with barycenters \(z(s,t1) = (s+1/2,t1/2)\) and \(z(s,t) = (s+1/2, t+1/2)\). By \(n^i_{\mathrm{top}}\) we denote the unit normal vector to the plane spanned by the points y(s, t), \(y(s+1,t)\), and \(y_{\mathrm{top}}^i := y((s,t)+v_i),\) with direction \((y(s+1,t)  y(s,t)) \times (y_{\mathrm{top}}^i  y(s,t))\), where for convenience we set \(v_1 := d_1 = (1,1)\) and \(v_2 := (0,1)\). Analogously, we let \(n^i_{\mathrm{bot}}\) be the unit normal vector to the plane spanned by y(s, t), \(y(s+1,t)\), and \(y_{\mathrm{bot}}^i:= y((s+1,t)v_i)\) with direction \((y(s,t)  y(s+1,t)) \times (y_{\mathrm{bot}}^i  y(s+1,t))\), see Fig. 3B.
Then, for all \(s,t \in {\mathbb {Z}}\), the signed incidence angles along the diagonal \(d_i\) of horizontal bonds are given by
Making use of the introduced notation, we are now in the position of formulating our first result. This is a simplified version of the later Theorem 2.8 and provides necessary conditions on the existence of admissible configurations.
Theorem 2.2
(Basic structure of admissible configurations) There exists \(\gamma ^* \in (0,\pi )\), depending only on \(\theta \), such that for every admissible configuration \(y:{\mathbb {Z}}^2\rightarrow {\mathbb {R}}^3\), possibly up to reorientation of the reference lattice, the following holds true:

(Constant form function along \(d_1\)) We have \(\tau (s,t) = \tau (s+1, t+1)\) for all \(s,t \in {\mathbb {Z}}\).

(Vanishing incidence angle along \(d_1\)) We have \(\gamma _1(s+1/2,t) = 0 = \gamma _1(s,t+1/2)\) for all \(s,t\in {\mathbb {Z}}\).

(Incidence angle along \(d_2\)) It holds that \(\gamma _2(s,t) = \gamma _2( s+1/2,t+1/2) \in \{ \pm \gamma ^*, 0\}\) for all \(s,t\in \frac{1}{2}{\mathbb {Z}}\) with \(s+t \in {\mathbb {Z}}+1/2\).
This theorem implies that ground states are essentially onedimensional, in the sense that they can be characterized as twodimensional deformations of onedimensional chains, see Fig. 4. Indeed, due to \(\tau \) being constant along \(d_1\), any cross section along \(d_2\) contains the same information. In particular, admissible configurations can be any combination of flat, rolledup/down areas in relation to the fact that the incidence angle along \(d_2\) can be 0 (flat areas), \(\gamma ^*\) (rolledup areas) or \(+\gamma ^*\) (rolleddown areas).
In the next subsections, we will present a refined version of Theorem 2.2, namely Theorem 2.8. We will show that Theorem 2.8 below implies Theorem 2.2. In Sect. 3 we then prove Theorem 2.8, which then also implies Theorem 2.2.
2.3 Geometry of Optimal Cells and Construction of 4Tiles
We aim at obtaining a complete characterization of admissible configurations, by resorting to socalled 4tiles. To introduce this concept, we first need to investigate the geometry of optimal cells in more detail. First, we consider an admissible deformation y and an optimal cell of the configuration, consisting of the points \(y_1,\ldots ,y_4\) and the corresponding midpoints \(m_1 = (y_1 + y_3)/2\) and \(m_2 = (y_2 + y_4)/2\), as indicated in Fig. 2. We denote the length of the diagonal by \(2v:= \vert y_1  y_3 \vert = \vert y_2  y_4 \vert \). By the cosine rule we have
Setting \(d := \vert y_1  m_2 \vert = \vert y_3  m_2 \vert = \vert y_2  m_1 \vert = \vert y_4  m_1 \vert \), we obtain by Pythagoras’ theorem \(d = \sqrt{1v^2}= \sqrt{(1+\cos \theta )/2}\). This allows us to calculate the kink angle \(\kappa ^*\) of an optimal cell by
with \(h = \sqrt{12v^2}\), see also Lemma 2.5. We refer to Fig. 5 with the optimal cell formed by \(\lbrace C,M_2,E_2,M_3 \rbrace \) for an illustration. For \(\theta = \pi /2\) we have \(v/d = 1\), and thus \(\kappa ^* = \pi \). In this case, as expected, optimal cells are flat. Let us firstly observe that an optimal cell is uniquely determined by the coordinates of three points and the choice of the cell form.
Lemma 2.3
(Optimal cell). Given any three points \(y_1, y_2, y_4 \in {\mathbb {R}}^3\) of an optimal cell, i.e., points satisfying \(\vert y_1  y_4 \vert = \vert y_1  y_2 \vert = 1\) and \(\measuredangle y_4 y_1 y_2 = \theta \), there exists a unique fourth point \(y^\diagdown _3\) and \(y_3^\diagup \), respectively, such that \(\{y_1, y_2, y_3^\diagdown , y_4\}\) is optimal of form \(\diagdown \) and \(\{y_1, y_2, y_3^\diagup , y_4\}\) is optimal of form \(\diagup \).
For the proof, we refer to Sect. A.1. A priori, by prescribing only the common angle \(\theta \), many configurations are conceivable as each optimal cell can be of form \(\diagdown \) or form \(\diagup \), and neighboring cells can in principle be attached to each other with an arbitrary incidence angle. Condition (2.14) is therefore essential to reduce the number of admissible deformations. To take (2.14) into account, we now consider subconfigurations consisting of four optimal cells which are arranged in a square sharing one common point. Such structures are called 4tiles, and we refer to Fig. 5 for an illustration.
The point shared by all four optimal cells is called center and is denoted by C. The additional four points shared by two optimal cells are called middle points (as they are in the middle of the boundary of the 4tile), are denoted by \(M_i\) for \(i = 1,\dots ,4\), and are labeled counterclockwise such that
By construction, we have \(\measuredangle M_i \, C \, M_{i+1} = \theta < \pi /2\) which implies that the five points C and \((M_i)_{i=1}^4\) cannot be coplanar. We introduce the nonplanarity angles \(\delta _{13}\) and \(\delta _{24}\) by
Note that \(\delta _{13} \pi \) and \(\delta _{24} \pi \) indicate how far the five points C and \((M_i)_{i=1}^4\) are from being coplanar, and again refer to Fig. 5 for an illustration. The nonplanarity angles \(\delta _{13}\) and \(\delta _{24}\) are related by the following lemma.
Lemma 2.4
(Nonplanarity angles). The nonplanarity angles \(\delta _{13}\) and \(\delta _{24}\) satisfy
In particular, \(\delta _{13}\) and \(\delta _{24}\) coincide if and only if
Indeed, by (2.14) we always have \(\delta _{13} = \delta _{24}\) for every 4tile of an admissible configurations since \(M_1,\, C,\, M_3\) and \(M_2, \, C,\, M_4\) fulfill the condition in (2.14). This yields that \(\delta _\theta = 2\arccos (\sqrt{\cos \theta })\) is solely determined by \(\theta \). The proof relies on the geometry of optimal cells, i.e., on assumptions (2.12) and (2.13), and will be given in Sect. A.1.
We denote the four corner points of the 4tile by \(E_i\), \(i = 1,\dots ,4\), as indicated in Fig. 5. For the classification of all different 4tiles, it is convenient to frame 4tiles in a reference position, as given in the following proposition.
Lemma 2.5
(Reference position). (i) By applying a suitable isometry, every 4tile can be positioned in such a way that the center C coincides with the origin, and we have
where \(s = \sqrt{2}v\) (see (2.16)), \(h = \sqrt{12v^2}\), and \(\varsigma \in \lbrace 1,1 \rbrace \).
(ii) Fixing \(\varsigma \in \lbrace 1,1 \rbrace \), and the form of each of the four optimal cells, the positions of \((M_i)_{i=1}^4\) and \((E_i)_{i=1}^4\) are uniquely determined, up to isometry.
For the proof, we again refer to Sect. A.1. Lemma 2.5 entails that the middle points \((M_i)_{i = 1}^4\) are coplanar. For this reason, we call 4tiles coplanar in the following. By (2.14) coplanarity is a necessary condition for the admissibility of 4tiles.
In view of Lemma 2.5(ii), there are 32 different types of 4tiles. Indeed, there are \(2^4 = 16\) possibilities to distribute either a form \(\diagdown \) or a form \(\diagup \) optimal cell to the four positions of a 4tile. Additionally, one can do this construction for \(\varsigma =1\) or \(\varsigma =1\). As we show next, the different types can be classified into six classes which are invariant under rotation by \(\pi /2\) and reflection along the \(e_1\)\(e_2\)plane, see Table 1. A representative of each class is shown in Fig. 6. The names of the classes are inspired by their geometry: the Itile is intermediate between the zigzagshaped Ztile and the diagonally rolledup Dtile (cf. the example in (2.22)). Similarly, the Jtile joins the arrowheadshaped Atile with the Etile, whose periodic pattern resembles to egg cartons.
To denote a 4tile we use a matrixlike notation, where the form of the optimal cell in the square is represented by \(\diagdown \) or \(\diagup \) in the respective position in the matrix. The case of \(\varsigma =1\) is indicated with a \(+\)symbol in the center of the matrix, and \(\varsigma =1 \) is denoted with a −symbol. We use this notation since, given a 4tile in reference position, we have that for \(i=1,\ldots ,4\) the center satisfies \((C  M_i) \cdot e_3 > 0\) if \(\varsigma =1\) (e.g. in Fig. 6D) and \((C  M_i) \cdot e_3 < 0\) if \(\varsigma =1\) (e.g. in Fig. 5), see Lemma 2.5(i).
Reflection of a 4tile in reference position with respect to the \(e_1\)\(e_2\)plane interchanges the index \(+\) with −. Moreover, \(\diagup \) and \(\diagdown \) are exchanged, as observed in Sect. 2.2. Also a rotation by \(\pi /2\) interchanges the forms of the optimal cells, i.e., swaps \(\diagdown \) and \(\diagup \), see again Sect. 2.2. In addition, note that, by applying a \(\pi /2\) rotation, one needs to permute the entries of the matrix accordingly, e.g.,
for a clockwise rotation of the entries.
As an example, rotation leaves the 4tile invariant, as interchanging \(\diagup \) and \(\diagdown \) yields and the rotation of the entries then leads to . However, rotating clockwise, i.e., first swapping \(\diagdown \) and \(\diagup \) to obtain and then rotating the entries to , yields an 4tile of the same class, but with different type, see Table 1.
2.4 Boundary Orientation and Boundary Angles
In this subsection, we further refine the characterization of 4tiles by introducing a notion of boundary orientation. To this end, consider a 4tile with notation as indicated in Fig. 5, placed in reference position. We call three points \(E_{i1}\), \(M_i,\) and \(E_i\), and the two bonds in between a boundary of the 4tile, where the indices have again to be understood modulo 4. We define the boundary orientation of \(E_{i1} \, M_i \, E_i\) by
and the corresponding boundary angle by
Intuitively, the orientation describes the fact that the boundary points upwards (orientation \(\wedge \)) or downwards (orientation \(\vee \)), see Fig. 6 for an illustration. Boundary orientation and boundary angle are crucial for classifying admissible configurations as they provide compatibility conditions for neighboring 4tiles. To formalize this, we now introduce the notion of attached 4tiles.
Given two 4tiles T and \({{\tilde{T}}}\) with centers C and \({{\tilde{C}}}\), we say that the 4tiles are attached to each other if \(y^{1}(C)  y^{1}({{\tilde{C}}}) \in \lbrace 2 e_1,  2 e_1, 2 e_2 , 2 e_2 \rbrace \). Note that T and \({{\tilde{T}}}\) share exactly one of the middle points \((M_i)_{i=1}^4\) and \(({{\tilde{M}}}_i)_{i=1}^4\) (and the adjacent two corner points). This shared middle point is the center of the socalled middle 4tile which is formed by two optimal cells of T and two optimal cells of \({{\tilde{T}}}\).
The following result will be a key tool for the classification of admissible configurations.
Lemma 2.6
(Attachment of two 4tiles) If two 4tiles are attached to each other, the boundary angles and the boundary orientation at the shared boundary coincide. If the boundary orientation is \(\wedge \), the corresponding middle 4tile satisfies \(\varsigma = 1\) (see Lemma 2.5(i)), otherwise we have \(\varsigma = 1\).
Lemma 2.6 will be proved in Sect. A.2. The statement delivers necessary conditions for attaching two 4tiles. In fact, a crucial idea for proving the main theorem, Theorem 2.8, is excluding many situations by checking that boundary angles or boundary orientations do not match. In particular, this reasoning will allow us to prove that admissible configurations exclusively contain Z, D, and Itiles. To ease the readability, from now on we include the boundary orientation in the notation, at least for the relevant tiles, i.e., the Z, D, and Itiles. This allows for an easy check whether the boundary orientations match or not.
On lateral boundaries, we denote boundaries with orientation \(\wedge \) by <. Likewise, lateral boundaries with boundary orientation \(\vee \) are indicated by >. Table 2 gives an overview of admissible 4tiles with the new notation.
In the notation, we also denote corner points pointing downwards with \(\circ \) and corner points pointing upwards with \(\bullet \) (of course, always assuming that the 4tile is in reference position). As an example, we refer to (b) and (f) in Fig. 6 for and , respectively. Note that this notation is not part of the characterization of types, but is included only to visualize the directions along which the boundary rolls up or down, respectively. (In fact, a \(+\) in the center along with \(\diagup \) pointing towards \(+\) yields \(\circ \) in the corresponding corner. In a similar fashion, a − in the center along with \(\diagup \) not pointing towards − yields \(\bullet \).) This notation facilitates to determine the class of the 4tile as Ztiles have no \(\bullet \)/\(\circ \), Dtiles have two, and Itiles have exactly one.
Lemma 2.7
(Boundary orientations). The boundary orientations of the different boundaries of the Z, D, and Itiles are given as indicated in Table 2.
Lemma 2.7 will be proved in Sect. A.2. We close this subsection with an example illustrating Lemma 2.6. Let us attach the Ztile and the Dtile . From the notation we can directly see that by attaching via
the boundary orientation match at the shared boundary, i.e., the 4tiles can be attached to each other provided that also the boundary angles coincide. (This indeed holds true, as we will see later in Lemma 3.1.) The type of the middle 4tile can be determined directly by considering the forms of the four optimal cells in the middle, i.e., . As the shared boundary has orientation >, which corresponds to \(\vee \), Lemma 2.6 implies that the middle 4tile satisfies \(\varsigma = 1\). The latter implies a −symbol in the middle of the matrix, see the discussion below Lemma 2.5. Therefore, the middle 4tile is the Itile . Clearly, the procedure applies to all combinations of 4tiles.
2.5 Main Result: Characterization in Terms of 4Tiles
After having introduced the necessary notation and concepts in the previous subsections, we are ready to formulate our main result on the characterization of admissible configurations in terms of 4tiles. To this end, we need a variant of the form function, the socalled type functions: consider an admissible deformation y and let \(S_1 = 0\), \(S_2 = (1,0)\), \(S_3=(0,1)\), and \(S_4 = (1,1)\). For \(i=1,\ldots ,4\), we let \(\sigma _i\) be the function defined on \(2{\mathbb {Z}}^2\) such that \(\sigma _i(k,l)\) for \((k,l) \in 2{\mathbb {Z}}^2\) indicates the type of the 4cell with center \(y(S_i + (k,l))\). The four different functions account for the fact that a translation of \({\mathbb {Z}}^2\) by (0, 0), (1, 0), (0, 1), or (1, 1) leaves the deformed configuration invariant, but groups together different optimal cells to form 4tiles.With this definition at hand, we now state the main result of this paper.
Theorem 2.8
(Characterization of all admissible configurations) A deformation y is admissible if and only if, possibly up to rotation of the lattice \({\mathbb {Z}}^2\) by \(\pi /2\), the following holds true:
Only particular types of Z, D, and Itiles are admissible, namely, for \(i=1,\ldots ,4\) we have
Moreover, the type function is constant along \(d_1\), i.e., \(\sigma _i(s,t) = \sigma _i(s+2, t+2)\) for all \(s,t \in 2{\mathbb {Z}}\) and the following matching conditions are satisfied:

(M1)
for all \(s,t \in 2{\mathbb {Z}}\) we have

(M2)
for all \(s,t \in 2{\mathbb {Z}}\) we have
The theorem gives a complete characterization of all admissible configurations. First, it shows that only Z, D, and Itiles are admissible. More precisely, we see that only such D, and Itiles from Table 2 are admissible, which rollup/down along the same diagonal, and that the type function is constant along the other diagonal. In particular, no change between the direction of rollingup/down is admissible. This observation allows for a clear geometric interpretation: Ztiles correspond to flat areas and Dtiles induce rolledup/down areas. In order to match such 4tiles, the Itile arises naturally as a combination of the Ztile and Dtile. (See, e.g., Fig. 6B, which is a Dtile left and a Ztile right. See also the example in (2.22).) Clearly, rollingup/down exclusively along the other diagonal is admissible as well, corresponding exactly to the other collection of D, and Itiles from Table 2. However, after a rotation of the lattice \({\mathbb {Z}}^2\) by \(\pi /2\), one can always reduce to (2.23). Eventually, the matching conditions (M1) and (M2) further restrict the admissible combination of 4tiles, and account for the fact that the boundary orientations at shared boundaries of two attached 4tiles need to match, see Lemma 2.6. We close this discussion by noting that the characterization cannot be simplified further, i.e., there are indeed admissible configurations y which contain all eight types given in (2.23).
Let us now stress that Theorem 2.8 implies Theorem 2.2. To see this, we observe that the type functions \(\sigma _i\), \(i=1,\ldots ,4\), are constant along the diagonal \(d_1\). This along with the fact that all types in (2.23) have the same form of optimal cell (\(\diagdown \) or \(\diagup \)) along the diagonal \(d_1\) (i.e., in the lower left and upper right entry) shows that the form function \(\tau \) introduced in Sect. 2.2 satisfies \(\tau (s,t) = \tau (s+1, t+1)\) for all \(s,t \in {\mathbb {Z}}\).
The fact that all incidence angles along \(d_1\) vanish and that all incidence angles along \(d_2\) lie in \(\lbrace 0,\gamma ^*,\gamma ^*\rbrace \) (with the property that the value is constant along \(d_1\)) follows by an elementary computation. We defer the exact calculation to Appendix A.4. At this stage, we only mention that inside Ztiles, all incidence angles along both diagonals are equal to zero. On the other hand, for the Dtile the incidence angle along \(d_2\) is \(\gamma ^*\) and for it is \(\gamma ^*\). Itiles have incidence angles 0 and \(\pm \gamma ^*\), where the sign depends on \(\bullet \) or \(\circ \) in the notation.
3 The Proof of the Main Theorem
This section is devoted to the proof of Theorem 2.8. This hinges on two facts, namely, that (1) attaching two 4tiles is only possible if the boundary orientation at shared boundaries match and (2) that such attachment needs to lead to an admissible, i.e., coplanar middle 4tile. Firstly, we use these ideas to show that actually only Z, D, and Itiles are admissible, see Proposition 3.2. In a second step, we further restrict the set of admissible types by showing that D and Itiles necessarily need to rollup/down along the same diagonal, see Proposition 3.3. This is achieved by considering four 4tiles arranged in a square and exploiting the aforementioned compatibility conditions. With similar techniques, we subsequently show that along one diagonal the type has to be constant, see Proposition 3.4. Eventually, we provide another auxiliary result (Proposition 3.5) stating that four 4tiles arranged in a square can be indeed realized by an admissible configuration y if all compatibility conditions, including the matching conditions stated in Theorem 2.8, are satisfied. With these results at hand, we are then able to prove Theorem 2.8.
3.1 Admissible Classes of 4Tiles
In this subsection, we show that admissible configurations contain only Z, D, and Itiles and that pairs of such tiles can be attached. This is achieved in two steps. We start by calculating the different boundary angles introduced in (2.21). Then, by discussing the possibility of attaching two 4tiles along a boundary with the same boundary angle and the same boundary orientation, see (2.20), we are able to show that Z, D, and Itiles are admissible, while E, A, and Jtiles are not.
We start by observing that there are exactly three different boundary types. In view of Lemma 2.5, we see that the three points forming a boundary (e.g., \(E_{i1}\), \(M_i\), and \(E_i\), see Fig. 5) are completely characterized by \(\varsigma \in \lbrace 1,1\rbrace \) and the form, i.e., form \(\diagdown \) or form \(\diagup \), of the two optimal cells adjacent to the boundary. (Strictly speaking, in Lemma 2.5(ii), this was only shown once the forms of all four optimal cells are fixed, but the argument clearly localizes at each boundary.)
This leads to at most \(2^3 = 8\) different boundary types, as indicated in Table 3. Given a 4tile in reference position, the boundary type remains invariant under reflection of the 4tile along the \(e_1\)\(e_2\)plane and the \(e_2\)\(e_3\)plane. This shows that the number of different boundary types reduces to three. We indicate the corresponding boundaries as Z, D, and Eboundaries, respectively, as the corresponding 4tiles have exclusively such boundaries, compare also Table 3 with Table 1. We also mention that Itiles have both Z and Dboundaries, but no Eboundaries, and that J and Atiles contain Eboundaries.
Lemma 3.1
(Boundary angles) The Zboundary angle and Dboundary angle of coplanar 4tiles are given by \(\delta _\theta = 2\arccos \left( \sqrt{\cos \theta }\right) \). The Eboundary angle of coplanar 4tiles is strictly smaller than \(\delta _\theta \).
Proof
We start by considering the Zboundary angle. Without restriction we consider a Ztile in reference position with notation as indicated in Fig. 5, satisfying \(M_2 = (0,s,h)\) for \(s,h>0\), where s and h are given in Lemma 2.5. We observe that the isometry \(x = (x_1,x_2,x_3) \mapsto (x_1,x_2,x_3) + (0,s,h) \) maps \(M_1\) to \(E_1\), C to \(M_2\), and \(M_3\) to \(E_2\). This yields that the Zboundary angle coincides with \(\delta _\theta \), see (2.14) and (2.18). The fact that the Dboundary angle coincides with the Zboundary angle is postponed to Corollary A.3, and relies on the fact that two 4tiles with the respective boundaries can be attached to each other, cf. Lemma A.2.
Eventually, we show that the Eboundary angle is strictly smaller. To this end, we let \(E_1= (s,s,0)\), \(M_2= (0,s,h) \), \(E_2 = (s,s,0) \) be again the points of the Ztile considered above. The corresponding points of an Etile in reference position are denoted by \({{\tilde{E}}}_1\), \({{\tilde{M}}}_2\), and \({{\tilde{E}}}_2\). (They are obtained by changing the form of the optimal cells containing \(E_1\) and \(E_2\), respectively.) By simple geometric considerations we find
for some \(p,q>0\). One can check that \(q = {{\tilde{E}}}_1 \cdot e_3 = {{\tilde{E}}}_2 \cdot e_3 > 2h\), see Lemma A.1(iv) below. Given that \({{\tilde{E}}}_1  {{\tilde{M}}}_2 =  {{\tilde{E}}}_2  {{\tilde{M}}}_2 = 1\), the Eboundary angle is calculated by \(\arccos (({{\tilde{E}}}_1  {{\tilde{M}}}_2) \cdot ({{\tilde{E}}}_2  {{\tilde{M}}}_2))\). We now compute by using (3.1) and \(q > 2h\) that
As \(\delta _\theta = \arccos (({E}_1  {M}_2) \cdot ( {E}_2  {M}_2))\) and \(\arccos \) is strictly decreasing on \([1,1]\) we find that the Eboundary angle is smaller than \(\delta _\theta \). This concludes the proof. \(\square \)
Proposition 3.2
(Nonadmissible classes of 4tiles). An admissible configuration does not contain E, A, and Jtiles.
Proof
Suppose by contradiction that the configuration contains a 4tile of class E, A, or J. As each E, A, or Jtile contains at least one Eboundary, see Tables 1 and 3, by Lemma 2.6 and Lemma 3.1 we deduce that the configuration contains at least two adjacent 4tiles in these three classes such that the shared boundary has an Eboundary angle. For the corresponding middle 4tile between the two 4tiles we thus get that the corresponding \(\delta _{13}\) or \(\delta _{24}\) as defined in (2.18) coincides with the Eboundary angle which is strictly smaller than \(\delta _\theta \) by Lemma 3.1. On the other hand, by (2.14) we have \(\delta _{13} = \delta _{24} =\delta _\theta \) for the nonplanarity angles of the middle tile, a contradiction. \(\square \)
3.2 Proof of the Main Result
In this subsection we give the proof of Theorem 2.8. The argument rests upon two propositions, showing that only certain arrangements of Z, D, and Itiles are admissible. A third auxiliary result verifies that such arrangements are indeed admissible. We start by stating these results, whose proofs are postponed to the next subsections. Recall the notation of the 4tiles in Table 2.
Proposition 3.3
(Rollup/down along one diagonal). Consider any four adjacent 4tiles of class Z, D, or I of an admissible configuration arranged in a square. Then all D and Itiles locally rollup/down along the same diagonal, i.e., the type of the four 4tiles is either exclusively contained in \({\mathcal {A}}\) or exclusively contained in \({\mathcal {B}}\), where
and
Note that \({\mathcal {B}}\) can be obtained from \({\mathcal {A}}\) through a rotation of the reference lattice by \(\pi /2\), and vice versa. The proposition shows that locally only 4tiles which roll along the same diagonal can be attached to each other. The following result states that locally admissible configurations have the same type along one of the diagonals.
Proposition 3.4
(Arrangements along diagonals). Consider four adjacent 4tiles of an admissible configuration with types either in \({\mathcal {A}}\) or in \({\mathcal {B}}\), see (3.2)–(3.3), arranged in a square and denoted by
If the types are in \({\mathcal {A}}\), we have \({\mathfrak {B}} = {\mathfrak {D}}\), and if the types are in \({\mathcal {B}}\), we have \({\mathfrak {A}} = {\mathfrak {C}}\).
The previous two results yield restrictions for the arrangement of 4tiles in admissible configurations. The next result shows that such arrangements are indeed admissible.
Proposition 3.5
(Admissible arrangements of 4tiles).
(i) If two coplanar 4tiles in \({\mathcal {A}}\) are attached along a boundary with matching boundary orientation, the resulting middle 4tile is a coplanar 4tile in \({\mathcal {A}}\).
(ii) If four adjacent coplanar 4tiles with types in \({\mathcal {A}}\) are arranged as
such that \({\mathfrak {B}} = {\mathfrak {D}}\) and such that the four 4tiles satisfy the matching conditions (M1)–(M2) stated in Theorem 2.8, there exists an admissible deformation \(y:\lbrace 0,1,2,3,4\rbrace ^2 \rightarrow {\mathbb {R}}^3\) such that the 4tiles of \(y(\lbrace 0,1,2,3,4\rbrace ^2 )\) have the types indicated in (3.4).
A similar statement holds for 4tiles with types in \({\mathcal {B}}\) by rotation of the reference lattice by \(\pi /2\). We are now in a position to prove our main result.
Proof of Theorem 2.8
Step 1: \(\Rightarrow \). We recall the definition of \(\sigma _i\), \(i=1,\ldots ,4\), before the statement of Theorem 2.8. Without restriction we only consider \(\sigma _1\) in the following proof. By Proposition 3.2 we have that the configuration only contains Z, D, and Itiles.
We next show that all types are either in \({\mathcal {A}}\) or in \({\mathcal {B}}\), see (3.2)–(3.3), i.e., rolling up/down occurs at most along one diagonal. Assume by contradiction that there were two 4tiles rolling along different diagonals, i.e., \(T_1 \in {\mathcal {A}} \setminus {\mathcal {B}}\) and \(T_2 \in {\mathcal {B}} \setminus {\mathcal {A}}\). Choose \(s_i,t_i \in 2{\mathbb {Z}}\), \(i=1,2\), such that \( \sigma _1(s_1,t_1) = T_1\) and \(\sigma _1(s_2,t_2) = T_2\). By Proposition 3.3 we can apply Proposition 3.4 and thus find \(\sigma _1(s_1+r,t_1+r) = T_1\) and \(\sigma _1(s_2+r',t_2r') = T_2\) for all \(r,r' \in 2{\mathbb {Z}}\). For a particular choice of r and \(r'\) this entails \(T_1 = T_2\) or that \(T_1\) is adjacent to \(T_2\). In both cases, we obtain a contradiction to Proposition 3.3.
This shows that all types of 4tiles are either in \({\mathcal {A}}\) or \({\mathcal {B}}\). Up to a rotation of the reference lattice by \(\pi /2\), we may suppose that all types of 4tiles lie in \({\mathcal {A}}\), which corresponds to the notation of Theorem 2.8. By Proposition 3.4 we get that the type function is constant along \(d_1\), i.e., \(\sigma _i(s,t) = \sigma _i(s+2, t+2)\) for all \(s,t \in 2{\mathbb {Z}}\) and all \(i=1,\ldots ,4\).
It remains to show that the matching conditions (M1) and (M2) hold true as indicated in the statement. These properties rely on the fact that the boundary orientations of each two attaching 4tiles need to match, cf. Lemma 2.6.
We only prove matching condition (M1) as the proof for (M2) follows along similar lines. Since the type function is constant along \(d_1\), i.e., \(\sigma _1(s,t) = \sigma _1(s+2, t+2)\) (\(s,t \in 2{\mathbb {Z}}\)), for any \(s,t \in 2{\mathbb {Z}}\) such that , we have one of the two possibilities
where the boundaries of the 4tiles with type \(\sigma _1(s,t) = \sigma _1(s+2,t+2)\) are depicted with solid lines. The given boundary orientations and Lemma 2.6 imply that only a 4tile from (compare Table 2) can be attached in the blank position top left indicated by the dotted 4tile (where its straight boundaries represent arbitrary boundary orientations). Within the class of admissible 4tiles \({\mathcal {A}}\) in (3.2), exactly the four choices match this boundary orientation. Conversely, for \(s,t \in 2{\mathbb {Z}}\) such that an arrangement as above yields one of the two possibilities
However, due to the given boundary orientations, the 4tiles in
are the only 4tiles from \({\mathcal {A}}\) which can be attached in the blank position bottomright, again indicated with the dotted 4tile. This concludes the check of the matching conditions (M1).
Step 2: \(\Leftarrow \). The existence of an admissible configuration \(y:{\mathbb {Z}}^2 \rightarrow {\mathbb {R}}^3\) follows directly from Proposition 3.5(ii) and an induction argument. Indeed, (2.12) and (2.13) are satisfied since each cell is optimal. To see (2.14), it suffices to check that all 4tiles are coplanar. In fact, then (2.14) follows from Lemma 2.4. First, by construction in Proposition 3.5(ii) we get that all 4tiles related to the type function \(\sigma _1\) are coplanar. By using Proposition 3.5(i) we find that also the 4tiles related to the other type functions \(\sigma _i\), \(i=2,3,4\), are in \({\mathcal {A}}\) and are coplanar. This shows that all 4tiles are coplanar, as desired. \(\square \)
3.3 Rolling Along One Diagonal
This subsection is devoted to the proof of Proposition 3.3. The proof fundamentally relies on Lemma 2.6, i.e., the fact that the boundary orientations of attached 4tiles match. To this end, we will make extensive use of the matrix diagrams introduced in Table 2 in order to exclude certain arrangements of 4tiles. Unfortunately, not all nonadmissible cases can be ruled out by such compatibility analysis and we also need to consider some more refined tools, based on the real threedimensional geometry of the 4tiles. For this reason, we will use the following lemma concerning the attachment of four coplanar 4tiles. Recall the types of 4tiles \({\mathcal {A}}\) and \({\mathcal {B}}\) introduced in (3.2)–(3.3), as well as the different types of boundaries in Table 3.
Lemma 3.6
(Arrangements of four 4tiles) Consider four adjacent 4tiles of an admissible configuration with types either in \({\mathcal {A}}\) or in \({\mathcal {B}}\), see (3.2)–(3.3), arranged in a square and denoted by
Then: (i) If three tiles are Ztiles and one tile is an Dtile, then the Dtile is in \(\lbrace {\mathfrak {A}}, {\mathfrak {C}}\rbrace \) (case \({\mathcal {A}}\)) or in \(\lbrace {\mathfrak {B}}, {\mathfrak {D}}\rbrace \) (case \({\mathcal {B}}\)).
(ii) If two tiles are Ztiles and two tiles are Dtiles, then the Ztiles are arranged along one diagonal and the Dtiles along the other diagonal.
(iii) If three tiles are Dtiles and one tile is a Ztile, then the Ztile is in \(\lbrace {\mathfrak {A}}, {\mathfrak {C}}\rbrace \) (case \({\mathcal {A}}\)) or in \(\lbrace {\mathfrak {B}}, {\mathfrak {D}}\rbrace \) (case \({\mathcal {B}}\)).
(iv) The arrangement
is not admissible.
We postpone the proof of this lemma to Appendix A.3 and proceed with the proof of Proposition 3.3.
Proof of Proposition 3.3
We proceed in two steps: in Step 1 we show that two attached 4tiles cannot rollup/down along different diagonals. In Step 2 we show that in four adjacent 4tiles arranged in a square, the two pairs of diagonal 4tiles cannot rollup/down along different diagonals. These two steps imply the statement.
Step 1: Attached 4tiles. Up to interchanging the roles of \(\bullet \) and \(\circ \), and up to reflection along the \(e_1\) or the \(e_2\)axis, there are six different cases to address:
Here, the symbol is a placeholder both for the corresponding Itile and the Dtile . The meaning of the other symbols is analogous. For the proof, we refer the reader to Table 2 which collects all possible 4tiles.
Case 1: . This case leads to a contradiction to Proposition 3.2 as necessarily the middle 4tile is the Atile . As an example, among the four possibilities, we consider the case where both 4tiles are Itiles. In this case, we have .
Case 2: . This case ensues if two 4tiles with different boundary orientations are attached, which contradicts Lemma 2.6. As an example, among the four possibilities, we consider the case where both 4tiles are Dtiles. In this case, we have .
Case 3: . First, if both 4tiles are Dtiles, then up to a reflection along the \(e_2\)axis, we are in Case 1 and obtain a contradiction as explained before. In the case that one is a Dtile and the other is an Itile, we obtain a contradiction to Lemma 2.6 as then the boundary orientations do not match. In fact, these two last cases are and .
We can therefore assume that both 4tiles are Itiles, i.e., take the form . We will now consider which 4tiles are admissible on top of the given 4tiles. Since we have already ruled out Case 1 and the boundary orientations need to match by Lemma 2.6, we see that on top of the left Itile we can only have , , , , or , and on top of the right Itile we can only have , , , , or . In any case, the 4tile in the middle of the four considered 4tiles, will be an Atile of the form or . This contradicts Proposition 3.2 and concludes the proof of Case 3.
Case 4: . If both 4tiles are Dtiles, then up to a reflection along the \(e_2\)axis, we are in Case 2 and obtain a contradiction as explained before. If both 4tiles are Itiles, we have , i.e., the boundary orientations are different and we obtain a contradiction to Lemma 2.6. The two remaining possibilities are and . We prove the contradiction only for the first configuration as the second configuration can be treated along similar lines. In order to do so, we proceed as in Case 3 and attach 4tiles at the top, yielding
In (3.7), the straight dotted lines encompass all possible boundary orientations. We start by noting that the Itile in the middle of is of the form .
Since we have already ruled out Case 1 and the boundary orientations need to match by Lemma 2.6, only the 4tiles can be attached on top of the Dtile (left), i.e., at position \({\mathfrak {A}}\). Similarly, on top of the Itile (right) at position \({\mathfrak {B}}\) we can only attach the 4tiles , see Table 2. As the boundary orientations between \({\mathfrak {A}}\) and \({\mathfrak {B}}\) have to match as well, there are only eight possibilities of the upper two 4tiles which are indicated in the first two columns of Table 4. The two upper 4tiles \({\mathfrak {A}}\) and \({\mathfrak {B}}\) form middle 4tiles which are indicated in the third column of Table 4. Note that the 4tile attached on the bottom of this 4tile is exactly the middle 4tile between the original two 4tiles, i.e., . Therefore, in the first four cases we obtain a contradiction to Lemma 2.6 since the boundary orientations of the shared boundary of the two middle 4tiles do not match.
For the second four cases we need a different argument instead. To this end, we consider also the middle 4tile between the Dtile and \({\mathfrak {A}}\) (left middle 4tile) and the middle 4tile between the Itile and \({\mathfrak {B}}\) (right middle 4tile), see the last two columns in Table 4. We observe that in none of the cases the boundary orientations of the shared boundary of the left and right middle 4tiles match. This is again a contradiction to Lemma 2.6, concluding the check of Case 4.
Case 5: or : Without restriction we address only the first case as the second can be treated analogously (and, in fact, obtained by a rotation). We have to distinguish two cases. Firstly, the 4tile on the left is a Dtile, i.e., . Then up to a reflection along the \(e_2\)axis, we are in Case 1 and obtain a contradiction as explained before. Secondly, if the left 4tile is not a Dtile, it has to be an Itile. We obtain the two possible configurations and which both contradict Lemma 2.6 as the boundary orientations do not match.
Case 6: or . Without restriction we address only the first case as the second can be treated analogously. If the 4tile on the left is a Dtile, then we have the two possibilities and . Thus, the boundary orientations do not match which contradicts Lemma 2.6. If the right 4tile is a Dtile, we are in Case 4 and obtain a contradiction as explained before.
Therefore, both 4tiles have to be Itiles, i.e., we have
As in Case 4, we consider two 4tiles attached on the top. By using arguments similar to the ones above, we will show that the only possible choice how to assemble the four 4tiles would be given by
This, however, is excluded by Lemma 3.6(iv). To see (3.9), in view of the fact that we have already ruled out Cases 1–5 and the boundary orientations need to match by Lemma 2.6, only the 4tiles
can be attached on top of the left Itile in (3.8). Analogously, on top of the right Itile in (3.8) we can only attach the 4tiles
see Table 2. As in Case 4 we consider the middle 4tile between the left Itile in (3.8) and the 4tile on top of it (left middle 4tile) and the middle 4tile between the right Itile in (3.8) and the 4tile on top of it (right middle 4tile). In view of (3.10)–(3.11), there are only the cases indicated in Table 5. From Table 5 we see that the boundary orientations of the shared boundary of the two middle 4tiles can only match if the right middle 4tile is of type . By (3.11) this shows that only can be attached on top of the right Itile . Then, in view of (3.10), only can be attached on top of the left Itile as the other four 4tiles in (3.10) do not math the boundary orientation of . This shows that (3.9) holds, and concludes the proof of Case 6.
Step 2: 4tiles on the diagonal. We now show that in four adjacent 4tiles arranged in a square, the two pairs of diagonal 4tiles cannot rollup/down along different diagonals. Up to interchanging the roles of \(\bullet \) and \(\circ \), and up to reflection along the \(e_1\) or the \(e_2\)axis, there are two cases to consider, where Case 1 represents one of the eight situations
and Case 2 represents one of the eight situations
Here, as in Step 1, the symbols \(\bullet \) and \(\circ \) indicate both the corresponding Itile and Dtile. Without restriction we address only the first configuration in both cases as all other situations can be treated along similar lines.
Case 1. We start by introducing the labeling
We preliminarily note that, in view of Step 1, for \({\mathfrak {B}}\) and \({\mathfrak {D}}\) only 4tiles in \({\mathcal {A}} \cap {\mathcal {B}}\) are admissible, see (3.2)–(3.3), i.e., the two Ztiles and . We distinguish three different subcases:
Case 1.1. If \({\mathfrak {A}}\) is the unique Itile, then Lemma 2.6 for the boundary between \({\mathfrak {A}}\) and \({\mathfrak {D}}\) as well as the boundary between \({\mathfrak {D}}\) and \({\mathfrak {C}}\) implies that the 4tile \({\mathfrak {D}}\) cannot be a Ztile. In fact, the boundary orientation of \({\mathfrak {A}}\) on the right is \(\wedge \) (indicated by < in the notation) and the boundary orientation of \({\mathfrak {C}}\) on top is \(\vee \).
Case 1.2. By a similar reasoning, if \({\mathfrak {A}}\) is the unique Dtile and \({\mathfrak {C}}\) is the unique Itile, Lemma 2.6 implies that the 4tile \({\mathfrak {B}}\) cannot be a Ztile.
Case 1.3. If both \({\mathfrak {A}}\) and \({\mathfrak {C}}\) are Dtiles, we again use Lemma 2.6 and see that the 4tiles \({\mathfrak {B}}\) and \({\mathfrak {D}}\) can only be of type . Therefore, we need to consider the configuration
The middle 4tile between \({\mathfrak {A}}\) and \({\mathfrak {B}}\) is given by and the middle 4tile between \({\mathfrak {C}}\) and \({\mathfrak {D}}\) is given by . Their shared boundary have mismatching boundary orientations, contradicting Lemma 2.6.
Case 2. We start by introducing the labeling
As in Case 1, due to Step 1, for \({\mathfrak {B}}\) and \({\mathfrak {D}}\) only the two Ztiles and are admissible. We distinguish four different subcases:
Case 2.1. If both \({\mathfrak {A}}\) and \({\mathfrak {C}}\) are Itiles, Lemma 2.6 implies that the 4tile \({\mathfrak {B}}\) cannot be a Ztile.
Case 2.2. If both \({\mathfrak {A}}\) and \({\mathfrak {C}}\) are Dtiles, then Lemma 2.6 implies that the 4tile \({\mathfrak {B}}\) cannot be a Ztile.
Case 2.3. If \({\mathfrak {A}}\) is the unique Dtile and \({\mathfrak {C}}\) is the unique Itile, then Lemma 2.6 implies that the 4tile \({\mathfrak {D}}\) cannot be a Ztile.
Case 2.4. Now suppose that \({\mathfrak {A}}\) is the unique Itile and \({\mathfrak {C}}\) is the unique Dtile. Then \({\mathfrak {B}}\) and \({\mathfrak {D}}\) need to be of type . Therefore, we need to consider the configuration
and show that it is also not admissible. The Itile rolls up in direction top left, which has no influence in this (sub)configuration. In other words, by replacing in (3.12) the tile \({\mathfrak {A}}\) with the Ztile and showing that this modified configuration is not admissible, we also find that (3.12) is not admissible. In fact, in view of Lemma 3.6(i) and the fact that the Dtile lies in \({\mathcal {B}}\) (see (3.3)), we see that the modified version of (3.12) is not admissible. This concludes this step of the proof. \(\square \)
3.4 Constant Type Along the Diagonal
This subsection is devoted to the proof of Proposition 3.4.
Proof of Proposition 3.4
We assume without restriction that all four 4tiles lie in \({\mathcal {A}}\), see (3.2), as the other case is completely analogous. We consider
and note that we need to show that \({\mathfrak {B}}\) and \({\mathfrak {D}}\) are of the same type. We proceed in two steps: first, we show that \({\mathfrak {B}}\) and \({\mathfrak {D}}\) are of the same class, i.e., both have to be either Z, I, or Dtiles. In the second step, we then conclude that they even have to be of the same type. In the proof, we will use the following observation which directly follows from the definition of \({\mathcal {A}}\):
Step 1. In this step, we show that \({\mathfrak {B}}\) and \({\mathfrak {D}}\) are necessarily of the same class.
Case 1.1. If exactly one of the two tiles \({\mathfrak {B}}\) and \({\mathfrak {D}}\) is an Itile, in view of (3.13), we obtain a contradiction to Lemma 2.6 as not all boundary orientations of the four shared boundaries can match.
Thus, we can now assume that none of the tiles \({\mathfrak {B}}, {\mathfrak {D}}\) is an Itile. Actually, it is also not restrictive to assume that the tiles \({\mathfrak {A}}\) and \({\mathfrak {C}}\) are not of class I. Indeed, the upper left optimal cell of \({\mathfrak {A}}\) and the lower right optimal \({\mathfrak {C}}\) have no influence on the subsequent arguments in Cases 1.2–1.4 and can readily be replaced by the other type. This allows to replace tiles of class I by types of class Z or D in \({\mathcal {A}}\), without affecting the following arguments. Summarizing, it suffices to consider the case that all four 4tiles are Z or Dtiles.
Case 1.2. If three 4tiles are Dtiles and one tile is a Ztile, we only have that \({\mathfrak {B}}\) and \({\mathfrak {D}}\) are not of the same class if the Ztile lies in \(\lbrace {\mathfrak {B}}, {\mathfrak {D}} \rbrace \). This contradicts Lemma 3.6(iii).
Case 1.3. If three 4tiles are Ztiles and one tile is a Dtile, we only have that \({\mathfrak {B}}\) and \({\mathfrak {D}}\) are not of the same class if the Dtile lies in \(\lbrace {\mathfrak {B}}, {\mathfrak {D}} \rbrace \). This contradicts Lemma 3.6(i).
Case 1.4. If two 4tiles are of class Z and two of class D, the claim follows directly from Lemma 3.6(ii).
Step 2. In this second step we show that not only the class but also the type has to be constant along the diagonal. First, if we had different Ztiles or Dtiles along the diagonal, in view of (3.2), these two 4tiles would have different boundary orientations. Again by using (3.13), we obtain a contradiction to Lemma 2.6 as not all boundary orientations of the four shared boundaries can match.
We now address the case that \({\mathfrak {B}}\) and \({\mathfrak {D}}\) are Itiles. Again in view of (3.13) and the definition of \({\mathcal {A}}\), we find
since otherwise the boundary orientations do not match, contradicting Lemma 2.6. Whenever the type is not constant along the diagonal, the 4tile in the middle of the four 4tiles is an Atile which contradicts Proposition 3.2. For simplicity, we show this only in case a) as case b) follows along similar lines. In fact, by Lemma 2.6 we find that \({\mathfrak {A}}\) can only be of type , , , or , and \({\mathfrak {C}}\) can only be of type , , , or . Consequently, if \({\mathfrak {B}}\) is of type , in the middle we find the Atile or , and if \({\mathfrak {B}}\) is of type , we find the Atile or , see Table 1. \(\square \)
3.5 Admissible Arrangement of 4Tiles
This subsection is devoted to the proof of Proposition 3.5.
Proof of Proposition 3.5
Without restriction we perform the proof only for the types \({\mathcal {A}}\) defined in (3.2).
(i) We start by observing that each pair of 4tiles in \({\mathcal {A}}\) with matching boundary orientations can be attached since all boundary angles are either Z or Dboundary angles, see Table 3 and Table 1, and both angles coincide with \(\delta _\theta \), see Lemma 3.1. We first show that the 4tile in the middle is again in \({\mathcal {A}}\). In a second step, we check that the middle 4tile is also coplanar.
We recall that the type of the middle 4tile can by determined by considering the matrix notation, as exemplified in (2.22). In view of (3.2), we obtain the following six cases:
Case 1. Attaching two Ztiles, we find that the two tiles are of same type and the middle tile is the Ztile of the other type.
Case 2. Attaching two Dtiles, we find that the two tiles are of same type and the middle tile is again of this type.
Case 3. Attaching two Itiles, we can obtain all possible 4tiles in \({\mathcal {A}}\).
Case 4. Attaching a Z and a Dtile, we obtain an Itile in \({\mathcal {A}}\).
Case 5. Attaching a Z and an Itile, we obtain any Z and Itile in \({\mathcal {A}}\).
Case 6. Attaching a D and an Itile, we obtain any D and Itile in \({\mathcal {A}}\).
Note that in all cases above exactly 4tiles from \({\mathcal {A}}\) can occur, and no more than those.
It remains to show that the resulting middle 4tile is also coplanar. As attaching two 4tiles does not change the optimal angle \(\theta \), also the middle 4tile consists of four optimal cells with angle \(\theta \). Therefore, relation (2.19) holds for the middle 4tile as well. To conclude the proof, it suffices to show that one of the nonplanarity angles \(\delta _{13}\) and \(\delta _{24}\) of the middle 4tile is equal to \(\delta _\theta \). To this end, note that one of these angles coincides with the boundary angle of the shared boundary of the two 4tiles. By Lemma 3.1 this angle is equal to \(\delta _\theta \).
(ii) We proceed constructively to show that every configuration consisting of four 4tiles from \({\mathcal {A}}\) arranged in a square satisfying the matching conditions (M1)–(M2) is admissible, i.e., can be realized by an admissible deformation y. By assumption, \({\mathfrak {B}}\) and \({\mathfrak {D}}\) are of the same type. Then, one can check that, for any choice of \({\mathfrak {A}}, {\mathfrak {C}} \in {\mathcal {A}}\) satisfying the matching conditions (M1)–(M2), the boundary orientations of \({\mathfrak {A}}, {\mathfrak {C}}\) match with those of \({\mathfrak {B}}\) and \({\mathfrak {D}}\). In view of Lemma A.1(i), fixing \({\mathfrak {B}}\) in reference configuration and translating \({\mathfrak {D}}\) from its reference position by the vector (2s, 2s, 0), we see that these two 4tiles share exactly one corner point, and we have \(\vert P  {{\tilde{P}}} \vert = \vert Q  {{\tilde{Q}}} \vert = \sqrt{(2s)^2 + (2s)^2} = 4v\), where \(P, Q \in {\mathfrak {B}}\) and \({{\tilde{P}}}, {{\tilde{Q}}} \in {\mathfrak {D}}\) are the corner vertices indicated in Fig. 7. By Lemma A.1(i) the opposite corner points along the diagonal \(d_1\) have distance 4v, i.e., \(\vert E_1^{\mathfrak {A}}  E_3^{\mathfrak {A}} \vert = \vert E_1^{\mathfrak {C}}  E_3^{\mathfrak {C}} \vert = 4v\). Therefore, we can translate \({\mathfrak {A}}\) and \({\mathfrak {C}}\) from their reference positions such that their opposite corner points coincide with P and \({{\tilde{P}}}\) and Q and \({{\tilde{Q}}}\), respectively. Since, for every 4tile the distance between its center and a corner point equals \(\sqrt{s^2+s^2} = 2v\), see (2.16) and Lemma 2.5(i), after rotating \({\mathfrak {A}}\) and \({\mathfrak {C}}\) about \((0,2s,0) + {\mathbb {R}}(1,1,0)\) and \((2s,0,0) + {\mathbb {R}}(1,1,0)\), respectively, as indicated in Fig. 7, the corner points of \({\mathfrak {A}}\), \({\mathfrak {B}}\), \({\mathfrak {C}}\), and \({\mathfrak {D}}\) in the interior of the configuration coincide. As the boundary orientations match by (M1)–(M2) and the boundary angles coincide by Lemma 3.1, also the respective middle points coincide after rotation of \({\mathfrak {A}}\) and \({\mathfrak {C}}\). This along with part (i) of the statement shows that the configuration is indeed realizable by an admissible configuration \(y:\lbrace 0, 1,2,3,4\rbrace ^2 \rightarrow {\mathbb {R}}^3\). This concludes the proof. \(\square \)
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Acknowledgements
This work is partially supported by the FWFDFG grant I 4354, the FWF grants F 65, W 1245, I 5149, and P 32788, and the OeADWTZ project CZ 01/2021. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics–Geometry–Structure.
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Appendix A: Remaining Proofs
Appendix A: Remaining Proofs
1.1 Geometry of Optimal Cells and 4Tiles
This subsection is devoted to the proofs of the lemmas stated in Sect. 2.3.
Proof of Lemma 2.3
Recall that \(m_2 :=(y_2 + y_4)/2\) is the middle point between \(y_2\) and \(y_4\), cf. Fig. 2. We define \(a = m_2  y_1\), with \(\vert a \vert = d\). Let n be a normal vector to the plane spanned by \(y_1\), \(y_2\), and \(y_4\), in direction \((y_2  y_1) \times (y_4  y_1)\).
Observe that by assumption the fourth point \(y_3\) has to satisfy \(\vert y_2  y_3 \vert = \vert y_4  y_3 \vert = 1\) and thus has to lie on the plane spanned by a and n. Therefore, we can make the ansatz
where \(v_3\) and \(h_3\) are to be determined, see Fig. 8. Note that in ± we choose \(+\) for form \(\diagdown \) and − for form \(\diagup \) . To conclude, we are left to prove that \(v_3\) and \(h_3\) can be determined uniquely. Since the cell is optimal, we have \(\measuredangle y_3\, m_2\, y_1 = \kappa ^*\) (see (2.17)) as well as \(\vert a \vert = \vert m_2  y_1 \vert = m_2  y_3  = d\). Consequently, the triangle with vertices \( y_1\), \(m_2\), and \(y_3\) and thus also the values of \(v_3\) and \(h_3\) are uniquely determined. \(\square \)
For convenience, we proceed with the proof of Lemma 2.5 and show Lemma 2.4 afterwards.
Proof of Lemma 2.5
In the proof, we again use the notation indicated in Fig. 5. We recall the definition in (2.18) and drop for the moment the condition \(\delta _{13} = \delta _{24}\) induced by (2.14). To verify that every 4tile can be placed in reference position, we first rotate and translate the 4tile such that \(C=0\) and \(M_1 = (s_1,0,h_1)\), and \(M_3 = (s_1,0,h_1)\), where a simple trigonometric relation yields
Here, we note that \(s_1>0\), while \(h_1\) is negative whenever \(\delta _{13} > \pi \). We now show that the coordinates of \(M_2\) and \(M_4\) are given by
where \(s_2= \sin (\delta _{24}/2)\) and \(h_2 = \cos (\delta _{24}/2)\). We focus on \(M_2\) since the argument for \(M_4\) is analogous. For convenience, we write \(M_2 = (p_1,p_2,p_3)\) and use the definition of optimal cells, i.e., \(\measuredangle M_1 \, C \, M_2 = \theta = \measuredangle M_2 \, C \, M_3\) and \(M_1 = M_2 = M_3 = 1\), to find
By combining the two equalities we get \(p_1 = 0\). In view of (A.1), \(p_3\) is then given by
and, since \(M_2 = 1\), we find \(p_2 = \sqrt{1p_3^2}\). Thus, we have \(M_2 =(0,p_2,p_3)\). By a similar argument we find \(M_4 = (0,p_2,p_3)\). To conclude for (A.2), we need to find the relation between \(p_2\) and \(p_3\). To this end, we use the fact that \(\measuredangle M_2 \, C \, M_4 = \delta _{24}\) to calculate \(\cos (\delta _{24}) = M_2 \cdot M_4 = p_3^2  p_2^2\). This, together with \(p_2^2+p_3^2 = 1\), verifies that \(p_3 = \sqrt{( 1+\cos (\delta _{24}) )/2} = \cos (\delta _{24}/2)\) by using the doubleangle formula. Correspondingly, we find \(p_2 = \sin (\delta _{24}/2)\). This proves (A.2). Let us remark for later purposes that (A.3) implies
From the condition \(\delta _{13}=\delta _{24}\) we get that \(s=s_1=s_2\) and \(h=h_1=h_2 = \sqrt{1s^2}\). We also let \(\varsigma = \mathrm{sgn}(h_1) = \mathrm{sgn}(h_2)\). To conclude the proof of (i), it remains to check that \(s = \sqrt{2}v\), where v is defined in (2.16), i.e., is chosen in such a way that 2v indicates the length of a diagonal in an optimal cell. This length can indeed be expressed as \(M_iM_{i+1} = \sqrt{2}s \) for \(i=1,\ldots ,4\), which yields the desired relation.
We proceed with the proof of (ii). By fixing \(\theta \), the angle \(\delta _\theta \) is also determined and, by (i), also fixing \(\mathrm{sgn}(h_1)\) determines completely the positions of the points \((M_i)_{i=1}^4\). In view of Lemma 2.3, the positions of \((E_i)_{i=1}^4\) are determined as well, as soon as the forms of the four optimal cells are given. \(\square \)
Proof of Lemma 2.4
In the proof of Lemma 2.5 we have already verified (2.19), see (A.4). Consider \( f_\theta :[0,\pi ]\rightarrow {\mathbb {R}}\) defined by
As \(\cos \theta >0\), \( f_\theta \) is decreasing and thus has at most one fixed point. Hence, \( f_\theta \) has exactly one fixed point given by \(\delta _\theta = 2 \arccos (\sqrt{\cos \theta })\). This eventually shows that \(\delta _{13}\) and \(\delta _{24}\) coincide if and only if \(\delta _{13} = \delta _{24} = \delta _\theta \). \(\square \)
We close this subsection with an elementary observation. We again refer to the notation in Fig. 5.
Lemma A.1
(i) For any coplanar 4tile in \({\mathcal {A}}\) (cf. (3.2)) in reference position, see Lemma 2.5(i), we have \(E_1 = (s,s,0)\) and \(E_3= (s,s,0)\).
(ii) For any Ztile in \({\mathcal {A}}\) in reference position, we have \(E_2 = (s,s,0)\) and \(E_4= (s,s,0)\).
(iii) For any Dtile in \({\mathcal {A}}\) in reference position, we have \(E_2E_4 < 4v\), where v is given in (2.16).
(iv) Assume that an optimal cell \(\{y_1,\dots , y_4\}\) is positioned in such a way that \(e_3 \cdot y_1 = 0\) and \(e_3 \cdot y_2 = e_3 \cdot y_4 = h\). Then, depending on its form, we have \(e_3 \cdot y_3 = 0\) or \(e_3 \cdot y_3 > 2h\).
Similar statements as (ii)–(iv) hold for \({\mathcal {B}}\) in place of \({\mathcal {A}}\) by changing the roles of the diagonals.
Proof
Without restriction, we consider a 4tile in \({\mathcal {A}}\) in reference position such that \(\varsigma = 1\), cf. Lemma 2.5, as the other case only amounts to reflection along the \(e_1\)\(e_2\)plane. By Lemma 2.5(i) we have that \(M_1 = (s,0,h)\), \(M_2 = (0,s,h)\), \(M_3 = (s,0,h)\), and \(M_4 = (0,s,h)\). The optimal cells \(\{C, M_1, E_1, M_2 \}\) and \(\{C, M_3, E_3, M_4\}\) are of form \(\diagup \), see Fig. 2 and (3.2). Thus, by Lemma 2.3 we get \(E_1 = (s,s,0)\) and \(E_3 = (s,s,0)\). This shows (i). We now suppose that the 4tile is either of class Z or of class D, i.e., is of type or . Therefore, the two optimal cells \(\{C, M_2, E_2, M_3 \}\) and \(\{C, M_4, E_4, M_1\}\) are of form \(\diagup \) (Dtile) and of form \(\diagdown \) (Ztile), which yields to a cross section along the direction \((1,1)\) as indicated in Fig. 9. We now obtain
Indeed, for the Ztile this follows from Lemma 2.3. For the Dtile we use Thales’ intercept theorem instead, with reference to Fig. 9. In particular, this implies (ii). Then, as in the Zcell the distance of the diagonals is \(4v = 2\sqrt{2}s\), (A.5) and Fig. 9 show that in the Dcell we have \(\vert E_2^D  E_4^D \vert < 4v\). This implies (iii). Eventually, property (iv) follows from (A.5). \(\square \)
1.2 Boundary Orientations and Attachment of Two 4Tiles
This subsection is devoted to the proofs of Lemma 2.6 and Lemma 2.7.
Proof of Lemma 2.6
The statement for the boundary orientation and the boundary angle, defined in (2.20)–(2.21), respectively, follows from the fact that the notions are determined uniquely by the three points which are shared by the two 4tiles. More precisely, given any 4tile in reference position, by applying a rotation about the \(e_3\) axis composed with a further small rotation (depending on \(\theta \)), and a translation one can ensure that a boundary of the 4tile is contained in the \(e_2\)\(e_3\)plane and is symmetric with respect to the \(e_1\)\(e_3\)plane. Provided that \(\theta \) is small, one can check that this transformation does not change the inequality in (2.20). Clearly, each two 4tiles with the same boundary angles can be transformed in this fashion in order to be matched along the shared boundary.
Consider now two attached 4tiles positioned such that the middle 4tile is in reference position, in particular, the shared middle point of the boundary is the origin. If the boundary orientation of the shared boundary is \(\wedge \), then both shared corner vertices satisfy \(E_{i1} \cdot e_3, \, E_i \cdot e_3 < 0\), see (2.20), and thus for the middle 4tile we have \(\varsigma = 1\). An analogous argument applies if the boundary is \(\vee \). \(\square \)
Proof of Lemma 2.7
First, we note that, for any 4tile in reference position, reflection about the \(e_1\)\(e_2\)plane interchanges all boundary orientations since the reflection changes the sign of any \(e_3\)component. Moreover, rotation around \(e_3\) by \(\pi /2\) leaves the boundary orientation invariant. A rotation in the matrix notation therefore simply rotates the corresponding sides and interchanges \(\vee \) with > and \(\wedge \) with <. For example, rotating by \(\pi /2\) counterclockwise, yields . This entails that it is enough to check the boundary orientations for one representative of any class in Table 2.
First, by Lemma 2.5 and Lemma A.1 we get that the orientation of all boundaries of the coplanar Dtile is \(\vee \). Indeed, assume that the 4tile is in reference position and use the notation of Fig. 5. Then the corner vertices \(E_1 \cdot e_3 = E_3\cdot e_3 = 0 \) and \(M_i \cdot e_3 = h\) for \(i=1,\ldots ,4\). Moreover, the optimal cells \(\{C, M_2, E_2, M_3\}\) and \(\{C, M_4, E_4, M_1\}\) are positioned as in Lemma A.1(iv). Thus, we can conclude that the corner vertices \(E_2\) and \(E_4\) have \(e_3\)coordinate strictly larger than 2h and hence, in view of (2.20), we find that the boundary orientation is \(\vee \).
Consider the Ztile in reference position. In this case, the middle points satisfy \(M_i \cdot e_3 = h \), \(i = 1, \dots , 4\) and, in view of the forms of the four optimal cells, the corner points satisfy \(E_i \cdot e_3 = 0\), \(i = 1,\dots , 4\). Thus, all four boundaries have orientation \(\wedge \).
We observe that the above arguments actually only take into account the relative position of the two optimal cells adjacent to a boundary. Thus, one can repeat the arguments above for the Itiles. For instance, has two \(\vee \) boundaries top and left, i.e., adjacent to \(\bullet \) as in a Dtile, and two \(\wedge \) boundaries right and bottom, as in a Ztile. \(\square \)
1.3 Arrangements of Four 4Tiles
In this subsection we prove Lemma 3.6. We start by a result about the mutual position of two attached 4tiles. To this end, recall the types of 4tiles \({\mathcal {A}}\) and \({\mathcal {B}}\) introduced in (3.2)–(3.3), as well as the definition of s and h in Lemma 2.5(i).
Lemma A.2
Let T and \({{\tilde{T}}}\) be two attached Z, I, or Dtiles of an admissible configuration. Without restriction, up to applying an isometry, suppose that T is in reference position, and that the shared boundary consists of the three points \(E_1\), \(M_1\), \(E_{4}\) and \({{\tilde{E}}}_2\), \({{\tilde{M}}}_2\), \({{\tilde{E}}}_{3}\), respectively, referring to the notation in Fig. 5. We denote the reference position corresponding to \({{\tilde{T}}}\) by \({{\tilde{T}}}'\). We denote by \(R^{{\mathcal {A}}}_{\alpha }\) the counterclockwise rotation around the axis (1, 1, 0) by the angle \(\alpha \), and \(R^{{\mathcal {B}}}_{\alpha }\) denotes the counterclockwise rotation around the axis \((1,1,0)\) by the angle \(\alpha \).
(1) If the shared boundary is a Zboundary of T, and a Zboundary of \({{\tilde{T}}}\), then we have \({{\tilde{T}}} = (2s,0,0)+ {{\tilde{T}}}'\).
(2) If the shared boundary is a Dboundary of T and a Zboundary of \({{\tilde{T}}}\), we have
where \(\kappa \) is defined in (2.17), and \(\varsigma _T\) corresponding to T is given in Lemma 2.5.
(3) If the shared boundary is a Zboundary of T and a Dboundary of \({{\tilde{T}}}\), we have
where \(\varsigma _{{{\tilde{T}}}'}\) corresponds to \({{\tilde{T}}}'\).
The case of two shared Dboundaries is not addressed here as we will not need it in the sequel. We warn the reader that, in the applications below without further mentioning, we will apply isometries to the tiles in order to reduce the positions to the ones indicated in the lemma. We postpone the proof of Lemma A.2 to the end of this subsection, and proceed with the proof of Lemma 3.6.
Proof of Lemma 3.6
(i) Without restriction we suppose that the tiles lie in \({\mathcal {A}}\) and we suppose by contradiction that the Dtile is given by \({\mathfrak {D}}\). We assume that \({\mathfrak {B}}\) is given in reference position. Then, by Lemma A.2(1) we see that \({{\mathfrak {C}}}\) is in reference position shifted by (2s, 0, 0), and \({{\mathfrak {A}}}\) is in reference position shifted by (0, 2s, 0). By Lemma 2.5 this implies that the coordinates of the points Q and P, indicated with \(\square \) and respectively \(\star \) in Fig. 10A, are given by \(Q = (s,3s,0)\) and \(P = (3s,s,0)\), respectively. In particular, we have that \(PQ = 2\sqrt{2}s = 4v\), cf. (2.16) and Lemma 2.5(i), which corresponds to the length of the diagonal in \({\mathfrak {D}}\). For the Dtile \({\mathfrak {D}}\) in \({\mathcal {A}}\), however, having the rolling direction as given in Fig. 10A, cf. (3.2), the corresponding diagonal has length smaller than 4v by Lemma A.1(iii), a contradiction.
(ii) Without restriction we suppose that the tiles belong to \({\mathcal {A}}\) and we suppose by contradiction that the Ztiles are in \({\mathfrak {A}}\), \({\mathfrak {B}}\), and that the Dtiles are in \({\mathfrak {C}}\), \({\mathfrak {D}}\), as in Fig. 10B. We also assume that \({\mathfrak {B}}\) is given in reference position. By Lemma A.2(1) we see that \({\mathfrak {A}}\) is in reference position shifted by (0, 2s, 0), and thus the point Q, indicated by \(\square \), has coordinates \(Q = (s,3s,0)\). By Lemma A.2(3) the position of the tile \({\mathfrak {C}}\) is obtained by taking the tile in reference configuration, rotating around the axis (1, 1, 0) by the angle \(\pm 2\kappa \), and then by a shifting by (2s, 0, 0). As the corners where no rollup occurs are left invariant under the rotation, we find by Lemma A.1(i) that the point P, denoted by a \(\star \) in Fig. 10B, has coordinates (3s, s, 0). This implies \(PQ = 2\sqrt{2}s = 4v\). As in (i), this contradicts Lemma A.1(iii) since the length of the diagonal in the Dtile \({\mathfrak {D}}\) where the tile rollsup is less than 4v.
(iii) Again without restriction we assume that the tiles belong to \({\mathcal {A}}\) and we suppose by contradiction that the Ztile is in \({\mathfrak {B}}\), as in Fig. 10C. We assume that \({\mathfrak {B}}\) is given in reference position. By Lemma A.2(3) the position of \({\mathfrak {C}}\) is obtained by taking the tile in reference configuration, rotation around the axis (1, 1, 0) by the angle \(\pm 2\kappa \), and then by a shifting by (2s, 0, 0) (exactly in this order). As the corners where no rollup occurs are left invariant under the rotation, we find by Lemma A.1(i) that the point P, marked with \(\star \) in Fig. 10C, has coordinates \(P = (3s,s,0)\). In a similar fashion, the position of \({\mathfrak {A}}\) is obtained by taking the tile in reference configuration, rotating around the axis (1, 1, 0) by the angle \(\pm 2\kappa \), and then by a shifting by (0, 2s, 0). Lemma A.1(i) yields that the point Q, indicated with \(\square \) in Fig. 10C, has coordinates \(Q= (s,3s,0)\). This implies \(PQ = 2\sqrt{2}s = 4v\), which as in (i) and (ii) contradicts Lemma A.1(iii) since the length of the diagonal in the Dtile \({\mathfrak {D}}\) where the tile rollup is less than 4v.
(iv) We finally show that (3.6) is not admissible. As before, we denote the 4tiles by \({\mathfrak {A}}, \ldots , {\mathfrak {D}}\), as indicated in (3.5). Our strategy hinges on (i)–(iii): we denote by \({{\tilde{Q}}}\) the right middle point of \({\mathfrak {A}}\) and by \({{\tilde{P}}}\) the upper middle point of \({\mathfrak {C}}\). In view of Lemma 2.5(i) applied on \({\mathfrak {D}}\), their distance necessarily needs to be \(\sqrt{2}s = 2v\). We will show, however, that this is impossible.
In order to do so, we first assume that \({\mathfrak {B}}\) is in reference position. In view of Lemma A.2(3), the position of the tile \({\mathfrak {C}}\) is obtained by taking the tile in reference configuration, rotating it around the axis \((1,1,0)\) by the angle \(2\kappa \), and then by a shifting by (2s, 0, 0) (exactly in this order). In a similar fashion, by Lemma A.2(2) the position of the tile \({\mathfrak {A}}\) is obtained by taking the tile in reference configuration, followed by a translation by (0, 2s, 0), and then by rotation around the axis (1, 1, 0) by the angle \(2\kappa \) (exactly in this order).
We will now change the coordinate system to simplify the notational realization of the procedure: we suppose that the common vertex of all three 4tiles lies in the origin and we reorient the coordinate system such that the rotation axis (1, 1, 0) coincides with \(e_1\) and the rotation axis \((1,1,0)\) with \(e_2\), see Fig. 11. Then, the points \({{\tilde{Q}}}\) and \({{\tilde{P}}}\) are given by \({{\tilde{Q}}} = {\mathcal {R}}^{e_1}_{2\kappa } Q\) and \({{\tilde{P}}} = {\mathcal {R}}^{e_2}_{2\kappa } P = {\mathcal {R}}^{e_2}_{2\kappa } P\), where \(Q = (v,v,h)\) and \(P = (v,v,h)\) are calculated by using Lemma 2.5, and the rotations are given by
An elementary calculation yields
where we used the definition of \(\kappa = \arctan (h/v)\), see (2.17), and the trigonometric identities \(\cos (2 \arctan (x)) = (1x^2)/(1+x^2)\) and \(\sin (2 \arctan (x)) = 2x/(1+x^2)\), as well as \((v,0,h) = {\mathcal {R}}^{e_2}_{2\kappa }(v,0,h)\). Consequently, we obtain
which is strictly larger than \((2v)^2\) since \(\kappa \in (0,\pi /2)\), see (2.17). This establishes a contradiction since, as stated above, the distance should be 2v. \(\square \)
Proof of Lemma A.2
(1) Since the shared boundary is a Zboundary of T and a Zboundary of \({{\tilde{T}}}\), and the boundary orientations of T and \({{\tilde{T}}}\) match at the shared boundary (see Lemma 2.6), Tables 2 and 3 show that \(\varsigma _T = \varsigma _{{{\tilde{T}}}}\) and that the middle 4tile between T and \({{\tilde{T}}}\), denoted by \(T_*\), is a Ztile. By Lemma 2.6 we also find \(\varsigma _{T_*} =  \varsigma _T\). Then by Lemma A.1(ii) it is elementary to check that \(C_{{{\tilde{T}}}}  C_T = (2s,0,0)\), where \(C_{{{\tilde{T}}}}\) and \(C_T\) denote the centers of the 4tiles, respectively.
(2) We prove the result only for the particular case of the two 4tiles and , as depicted in Fig. 12. In fact, the general case can be reduced to this situation by (a) replacing the optimal cells which are not adjacent to the shared boundary, as they do not affect the argument; and by (b) applying a suitable rotation or reflection.
Suppose that T is in reference position and denote the reference position of \({{\tilde{T}}}\) by \({{\tilde{T}}}'\). We define \({{\tilde{T}}}'':={{\tilde{T}}}' + (2s,0,0)\). In view of Lemma A.1(i),(ii), we see that T and \({{\tilde{T}}}''\) share exactly one corner point \(E_* = (s,s,0)\) as depicted in Fig. 12. Clearly, the rotation around the axis (1, 1, 0) by \(2\kappa \) leaves \(E_*\) invariant. We need to show that under this rotation the points \(P_i\), \(i=1,2\), are mapped to \({{\tilde{P}}}_i\), as depicted in Fig. 12. We denote by \(C_i\), \(i=1,2\), the two points on (1, 1, 0) which intersect the plane with normal vector (1, 1, 0) containing \(P_i\) and \({{\tilde{P}}}_{i}\). By Lemma 2.5 we find that \(C_1 = (s/2,s/2,0)\) and \(C_2 = 0 \). We need to check that
We first address \(i=1\). By Lemma 2.5, we have \(P_1 = (s,0,h)\) and \({{\tilde{P}}}_1 = (s,0,h)\). We also note that \(s = \sqrt{2}v\). This along with \(C_1 = (s/2,s/2,0)\), \(\kappa = \arctan (h/v)\) (see (2.17)), and the trigonometric identity \(\cos (2 \arctan (x)) = (1x^2)/(1+x^2)\) yields (A.6) by an elementary computation.
We now address \(i=2\). As \(C_2\) and \({{\tilde{P}}}_2\) form a diagonal of an optimal cell, see Fig. 12, by the definition before (2.16) we get \(C_2  {{\tilde{P}}}_2 = 2v = \sqrt{2}s\). On the other hand, by Lemma 2.5, we find \(P_2 = (s,s,0)\) and therefore \(P_2  C_2 = \sqrt{2}s\). This shows the first part of (A.6). To calculate the angle, we refer to the cross section in Fig. 13. Since this cross section is the one of an optimal cell in a rotated position, we can calculate the angle \(\measuredangle P_2 \, C_2 \, {{\tilde{P}}}_2\) using the definition of \(\kappa ^*\) in (2.17) and thus derive that \(\gamma = \kappa + (\pi  \kappa ^*)/2 = \kappa + \pi /2  (\pi  2\kappa )/2 = 2 \kappa \). This concludes the proof.
The proof of (3) is similar to (2) by interchanging the roles of the 4tiles. We omit the details. \(\square \)
We proceed with a simple consequence for boundary angles defined in (2.21).
Corollary A.3
The boundary angle of a Dboundary coincides the the boundary angle of a Zboundary.
This immediately follows from Lemma A.2(ii). Indeed, if the statement was not true, one could not attach the two 4tiles, as described in the previous proof.
1.4 Incidence Angles in Coplanar 4Tiles: Theorem 2.8 Implies Theorem 2.2
In this short subsection, we explain that Theorem 2.8 implies Theorem 2.2. In Sect. 2.5, we already addressed the type function. Therefore, it remains to consider the last two items in Theorem 2.2, i.e., the incidence angles defined in (2.15) between optimal cells.
Ztiles. We start by showing that the incidence angles between optimal cells in a Ztile along both diagonals are given by zero. Observe that reflection about the \(e_1\)\(e_2\)plane only interchanges the sign of the incidence angle. Thus, we assume without restriction that \(\varsigma = 1\) (cf. Lemma 2.5). Due to symmetry, it suffices to consider one of the four bonds contained in the 4tile, e.g., the bond connecting C and \(M_1\), referring to the notation in Fig. 5. Without restriction we only calculate the incidence angle along \(d_1\) as the other one can be calculated in a similar fashion, again exploiting symmetry. If the 4tile is in reference position, Lemma 2.5(i) implies that \(y_{\mathrm{top}}^1 = E_1 = (s,s,0)\), \(y_{\mathrm{bot}}^1 = M_4 = (0,s,h)\), \(C=0\), and \(M_1 = (s,0,h)\). Therefore, since \(n^1_{\mathrm{top}}\) is in direction \(M_1 \times E_1\) and \(n^1_{\mathrm{bot}}\) is in direction \(M_1 \times (M_4M_1)\), we find \(n^1_{\mathrm{top}} = n^1_{\mathrm{bot}} = \frac{1}{\sqrt{s^2+2h^2}}(h,h,s)\). Thus, in view of (2.15), we get that the incidence angle is \(\arccos (1) = 0\).
Dtiles. We now address a Dtile in (2.23), given in reference configuration. Due to symmetry, it is again not restrictive to consider only the bond connecting C and \(M_1\) and to suppose that \(\varsigma = 1\). Note that, due to Lemma 2.5 and Lemma A.1(i), we have \(E_1 = (s,s,0)\) and \(E_4\) satisfies \(E_4 \cdot e_3 = q \) and \(0< E_4 \cdot e_1 =  E_4 \cdot e_2 = p:= \sqrt{s^2q^2/2}< s\) for some \(q >0\). Since \(E_1\), \(M_1\), and \(M_4\) have the same position as in a Ztile, repeating the above calculation we find that the incidence angle along \(d_1\) is zero. We now consider the angle along \(d_2\). To this end, we first find that \( y_{\mathrm{top}}^2 = M_2 = (0,s,h) \) and \( y_{\mathrm{bot}}^2 = E_4 = (p,p,q)\), see Lemma 2.5. Therefore, since \(n^2_{\mathrm{top}}\) is in direction \(M_1 \times M_2\) and \(n^2_{\mathrm{bot}}\) is in direction \(M_1 \times (E_4M_1)\), we get \(n^2_{\mathrm{top}} = \frac{1}{\sqrt{s^2+2h^2}}(h,h,s)\), and an elementary computation yields \(n^2_{\mathrm{bot}} = v/v\) for \(v = (h,h,s) + (0, q s /p,0) \). This implies \(n^2_{\mathrm{top}}\) and \(n^2_{\mathrm{bot}}\) are not parallel and therefore by (2.15) the incidence angle, denoted by \(\gamma ^*\), is not zero.
To determine the sign of the nonzero incidence angle, we need to determine the sign of \( (y^2_{\mathrm{top}} y_{\mathrm{bot}}^2) \cdot (n^2_{\mathrm {top}}  n^2_{\mathrm{bot}}) = (y^2_{\mathrm{top}} y_{\mathrm{bot}}^2) \cdot n^2_{\mathrm {top}}  (y^2_{\mathrm{top}} y_{\mathrm{bot}}^2) \cdot n^2_{\mathrm{bot}}\). First note that \((y^2_{\mathrm{top}} y_{\mathrm{bot}}^2) = M_2  E_4 = (p, s+p, hq)\) which yields \((y^2_{\mathrm{top}} y_{\mathrm{bot}}^2) \cdot n^2_{\mathrm {top}} = \lambda qs\), with \(\lambda = 1/\sqrt{s^2 + 2h^2}\) and \((y^2_{\mathrm{top}} y_{\mathrm{bot}}^2) \cdot n^2_{\mathrm{bot}} = \mu qs^2/p \), with \(\mu = 1/\vert v \vert \). Therefore, we obtain \( (y^2_{\mathrm{top}} y_{\mathrm{bot}}^2) \cdot (n^2_{\mathrm {top}}  n^2_{\mathrm{bot}}) = \lambda qs  \mu qs^2/p < 0\). Hence, the incidence angle has a negative sign, see (2.15). Summarizing, we have shown that in Dtiles the angles are also zero along \(d_1\) and lie in \(\lbrace \gamma ^*,\gamma ^* \rbrace \) along \(d_2\).
Itiles. It is obvious that for Itiles, being combinations of Z and Dtiles, we find that the incidence angles along \(d_1\) are also 0 and along \(d_2\) they lie in \(\lbrace \gamma ^*,0, \gamma ^* \rbrace \).
We close the proof by the observation that, due to the symmetries in Z, D, and Itiles contained in \({\mathcal {A}}\), see (3.2), it is indeed elementary to check that \(\gamma _2(s,t) = \gamma _2( s+1/2,t+1/2)\) for all \(s,t\in \frac{1}{2}{\mathbb {Z}}\) with \(s+t \in {\mathbb {Z}}+1/2\).
1.5 Admissible Configurations and Ground States of the Energy
This subsection is devoted to the proof of Proposition 2.1.
Proof
Step 1. We start by introducing a specific unit cell: fix \(x_0 \in {\mathbb {Z}}^2\) and denote the four neighbors of \(x_0\) by \(x_1 = x_0 + e_1\), \(x_2 = x_0 + e_2\), \(x_3 = x_0e_1\), and \(x_4 = x_0e_2\). Given a deformation \(y:\lbrace x_0,\ldots ,x_4\rbrace \rightarrow {\mathbb {R}}^3\), we define
\(y_i = y(x_i)\) for \(i=0,\ldots ,4\), and we let \(y_{5} = y_{1}\). We introduce the cell energy by
where \(\theta _i = \measuredangle y_i\, y_{0} \, y_{i+1}\) for \(i=1,\ldots ,4\), as well as \(\delta _{13} = \measuredangle y_1 \, y_0\, y_3\) and \(\delta _{24} = \measuredangle y_2 \, y_0 \, y_4\). The cell \((y_i)_{i=0}^4\) is called optimal if it minimizes (A.7).
Let us start by relating the cell energy to the configurational energy in (2.1). To this end, let \(y:{\mathbb {Z}}^2 \rightarrow {\mathbb {R}}^3\) be a deformation, and for \(m \in {\mathbb {N}}\) let \(Q_m\) be the open square centered at 0 with sidelength 2m. For \(j \in {\mathbb {Z}}^2 \cap Q_m\) we denote by \(y^j = \lbrace y_0^j,\ldots , y^j_4 \rbrace \) the cell considered above for \(x_0 = j \). Then, in view of (2.1), owing to the fact that bonds related to nearestneighbors and nexttonearestneighbors are contained in two cells (apart from bonds intersecting \(\partial Q_m\)), whereas each bond angle is contained in exactly one cell, we find for every \(m \in {\mathbb {N}}\) that
Then, recalling the definition in (2.5) and by arguing as in [13, Proposition 2.1] we have that \(y:{\mathbb {Z}}^2 \rightarrow {\mathbb {R}}^3\) is a ground state if and only if for each \(x_0 \in {\mathbb {Z}}^2\) the corresponding cell \(\lbrace y_0,\ldots , y_4\rbrace \) is optimal. Note that there exist admissible configurations consisting of optimal cells by Theorem 2.8, e.g., a tiling with only Ztiles. Therefore, in the following it suffices to minimize the cell energy and to show that the unique minimizer is identified by having specific bond lengths and bond angles.
Step 2. Let \(\{y_0, y_1, y_2, y_3, y_4\}\) be an optimal cell. We show that \(\vert y_{j}  y_{0} \vert \in (1\eta , 1+ \eta )\) as well as \(\theta _j > \pi /2  \eta \) for \(j=1,\ldots ,4\). Assume first by contradiction that \(\vert y_{j}  y_{0}\vert \le 1\eta \) for some \(j = 1,\dots ,4\). Then by using \(v_2 \ge 1\), \(v_3 \ge 0\), the fact that \(v_2\) is decreasing on (0, 1), and (2.6) we get
In the last equation, we have also used that \(v_2(1) = 1\) and \(v_3(\pi ) = 0\). This estimate contradicts the optimality of the cell.
In a similar fashion, we assume by contradiction that there exists some bond angle \(\theta _j\), \(j=1,\ldots ,4\), such that \( \theta _j \le \pi /2  \eta \). Then, by \(v_2 \ge 1\), \(v_3 \ge 0\), and (2.8) we have
which is again in contradiction with the optimality of y.
We eventually show that for an optimal cell the bond lengths have to be less then \(1+\eta \). Basic trigonometry together with the least size of the bond lengths and bond angles ensures that secondneighbor bonds have at least length
where the last two inequalities hold for \(\eta \) sufficiently small. Assume now that \(y_j  y_{0} \ge 1 + \eta \) for some \(j=1,\dots , 4\). Then, we get by \(v_2 \ge 1\), \(v_3 \ge 0\), the fact that \(v_2\) increasing on \([1,\infty )\), and (2.7) that
The latter inequality once again contradicts optimality and we conclude that all firstneighbor bond lengths are at most \(1+\eta \).
Step 3. To simplify notation, we denote the collection of angles by \(\varvec{\theta } :=(\theta _i)_{i=1}^4 = (\theta _1,\dots ,\theta _4)\). We observe that \(\delta _{24}\) can be written as a function of \(\varvec{\theta }\) and \(\delta _{13}\), i.e., \(\delta _{24}= f( \varvec{\theta }, \delta _{13} )\), where the function f is explicitly given in Step 8, see (A.21). We will not need the exact form of this function, but only use that it is smooth for \(\theta _i\) in a left neighborhood of \(\pi /2\) and \(\delta _{13}\) in a small interval left of \(\pi \), see Step 8 below. Using Lemma 2.4 we find that in a cell with \(\theta _1=\cdots =\theta _4 = \theta \) it holds that \(\delta _{24} = f(\theta ,\ldots ,\theta ,\delta _{13}) = f_\theta (\delta _{13})\), where \(f_\theta (\delta ) := 2\arccos \left( \cos \theta /\cos (\delta /2)\right) \). Note that \(f_\theta \) has a unique fixed point \(\delta _\theta := 2 \arccos (\sqrt{\cos \theta })\). We decompose the cell energy \(E_{\mathrm{cell}}\) defined in (A.7) as
where \(\ell _i := y_iy_0\) for \(i=1,\ldots ,4\) and
We have proved that, if \(\{y_0,\dots ,y_4\}\) is optimal, firstneighbor bond lengths \(\ell _i\) lie in \((1\eta ,1 + \eta )\) and bond angles \(\theta _i\) lie in \((\pi /2  \eta , \pi ]\). Therefore, by using the convexity assumption (2.9) on F we find
where
Note that the inequality in (A.10) is strict whenever \(\ell _i\not = {\bar{\ell }}\) or \(\theta _i \not = {\bar{\theta }}\) for some \(i=1,\dots ,4\).
Step 4. We check that the map \((\ell ,\theta ) \mapsto F(\ell ,\ell ,\theta )\) is minimized on \((1\eta ,1+\eta ) \times (\pi /2\eta ,\pi ] \) at some \(\ell ^* \le 1\) and \({\theta }^*<\pi /2\). If we had \(\ell ^* > 1\), one could reduce F by reducing \(\ell \), noting that \(v_2\) is increasing in \((1,\infty )\) and recalling (A.8). This, however, would contradict optimality. We now exclude \(\theta ^* \ge \pi /2\). Indeed, in this case we could decrease \(\theta ^*\) by \(0<{\tilde{\theta }} \ll 1\) and by a Taylor expansion we would get that F changes to first order by
By \(\ell \in (1\eta ,1]\) and (2.11) we get that the above term is negative, which contradicts minimality.
Step 5. Next, we show that for \({\bar{\theta }}\) defined in (A.11) it holds that \({\bar{\theta }} \le \pi /4 + \theta ^*/2\). We also establish a bound from below on \(\delta _{13}\) and \(\delta _{24}\). The argument is based on the observation that by (A.9), (A.10), and the definition of \(\delta _\theta \) we find that \(4F(\ell ,\ell ,{\theta }) + 2v_3(\delta _{\theta })\) is an upper bound for the minimal cell energy for \((\ell ,\theta ) \in (1\eta ,1+\eta ) \times (\pi /2\eta ,\pi ]\). By definition we have \(\delta _\theta \rightarrow \pi \) as \(\theta \rightarrow \pi /2\). Thus, in view of (2.10), the monotonicity of \(v_3\), and \(\theta ^* \ge \pi /2\eta \), we can choose \(\eta \) sufficiently small depending on \(v_3\) and find \(\lambda >0\) small such that \(v_3 \le \varepsilon \) on \([\pi \lambda ,\pi ]\), \(v_3' \le \varepsilon \) on \([\pi  2\lambda , \pi ]\), and
We also suppose that \(\varepsilon \) is chosen small enough depending on \(v_2\), \(v_3\), and \(\theta ^*\) such that
Now, we can suppose \(\delta _{13},\delta _{24} \ge \pi \lambda \) (recall that \(\delta _{24} = f(\varvec{\theta },\delta _{13})\)) since otherwise we get
by using (A.9), (A.10), and (A.12), which contradicts minimality. In a similar fashion, we can suppose that \({\bar{\theta }}\) in (A.11) satisfies \({\bar{\theta }} \le \frac{\pi }{4} + \frac{\theta ^*}{2}\) as otherwise \(E_{\mathrm{cell}} (y) > 4F({\ell }^*,{\ell }^*,{\theta }^*) + 2v_3(\delta _{\theta ^*})\) follows using (A.9), (A.10), (A.12), and (A.13).
Step 6. We are left with the case \(\delta _{13} \ge \pi \lambda \) and \(\theta _1 + \ldots + \theta _4 = 4 {\bar{\theta }} \le \pi + 2\theta ^* < 2\pi \). In this step, we show that
with equality only if \(\ell _i = {\bar{\ell }}\) and \(\theta _i = {\bar{\theta }}\) for \(i=1,\dots ,4\).
We start by noticing that \(\theta _1 + \ldots + \theta _4 <2\pi \) and \(\theta _i > \pi /2  \eta \) for \(i=1,\ldots ,4\) imply \(\theta _i < \pi /2 + 3\eta \) for \(i=1,\ldots ,4\). Therefore, the convexity estimate in (A.10) can be improved by using the strong convexity assumption (2.9) on F, and we find
for some \(\alpha >0\). Moreover, a simple geometric argument shows that \(\delta _{13} = \pi \) implies \(\theta _1 + \ldots + \theta _4 = 2\pi \), see Fig. 14. Therefore, by a continuity argument and \(\theta _1 + \ldots + \theta _4 \le \pi + 2\theta ^*\) we get that \(\delta _{13} \le \delta ^*\) for some \(\delta ^* < \pi \) only depending on \(\theta ^*\). Consequently, we need to consider the case that \(\delta _{13} \in [\pi  \lambda ,\delta ^*]\) and \(\theta _i\in (\frac{\pi }{2}\eta ,\frac{\pi }{2}+3\eta )\).
If \(\alpha \sum _{i = 1}^4 \theta _i  {{\bar{\theta }}}^2 \ge 2\varepsilon \), by (A.9), (A.12), and (A.15) we obtain a contradiction to minimality as \(v_3(\delta _{13}) + v_3(f(\varvec{\theta },\delta _{13})) + \alpha \sum _{i = 1}^4 \theta _i  {{\bar{\theta }}}^2 \ge 2\varepsilon > 2v_3(\delta _{{\theta }^*})\).
If \(\alpha \sum _{i = 1}^4 \theta _i  {{\bar{\theta }}}^2 < 2\varepsilon \), we now show that \(f_{{{\bar{\theta }}}}(\delta _{13})\) cannot be too far away from \(f(\varvec{\theta },\delta _{13})\). Eventually, this will allow us to deduce (A.14). By choosing \(\varepsilon \) small enough and recalling that \({{\bar{\theta }}}<\pi /2\), we get that \(\theta _i \le \pi /2\) for \(i=1,\ldots ,4\). By Taylor’s Theorem there exists \( {{\varvec{z}}}\in \{ t\varvec{\theta } + (1t) \varvec{{\bar{\theta }}} \, : \, t \in [0,1] \}\), where \(\bar{\varvec{\theta }} = ({\bar{\theta }}, \dots , {\bar{\theta }})\), such that
where we used that \(\tfrac{\partial }{\partial \theta _i} f(\varvec{{\bar{\theta }}}, \delta _{13}) = \tfrac{\partial }{\partial \theta _j} f(\varvec{{\bar{\theta }}}, \delta _{13}) \) and thus
In (A.16) we denoted the largest eigenvalue of the Hessian \(\nabla _{\varvec{\theta }}^2 f({{\varvec{z}}},\delta _{13})\) with \(\lambda _{\max }\). Using the Gershgorin circle theorem, we find \(\vert \lambda _{\max } \vert \le 4 c_f\) where we use that f is smooth for \(\theta _i \in I:=[\frac{\pi }{2}\eta ,\frac{\pi }{2}]\) and \(\delta _{13} \in [\pi  \lambda ,\delta ^*]\), and define
The proof of the smoothness of f is deferred to the end of the proof in Step 8. Therefore, we obtained
For \(\varepsilon \) small enough such that \( 4 c_f \varepsilon \le \alpha \lambda \), due to \(\alpha \sum _{i = 1}^4 \theta _i  {{\bar{\theta }}}^2 < 2\varepsilon \), we have
Hence \(f_{{{\bar{\theta }}}}(\delta _{13}) \ge \pi  2\lambda \) as \(f(\varvec{\theta },\delta _{13}) \ge \pi \lambda \) by Step 5. Therefore, as \(v_3' \le \varepsilon \) on \([\pi 2\lambda ,\pi ]\) and \(\lambda >0\) small, we obtain by (A.17)
Consequently, (A.14) holds by applying (A.15) and (A.18) to (A.9).
Step 7. We now conclude the proof by showing
where \(\delta _{{\bar{\theta }}} =2 \arccos (\sqrt{\cos {\bar{\theta }}})\), and that equality holds only if \(\ell _i = {\bar{\ell }}\) and \(\theta _i = {\bar{\theta }}\) for \(i=1,\dots ,4\). To this end, we further estimate (A.14) by claiming
with equality if and only if \(\delta _{13} = f_{{\bar{\theta }}}(\delta _{13})\). Computing \( g'(\delta ) = v_3'(\delta ) + v_3'(f_{{{\bar{\theta }}}}(\delta )) \, f'_{{{\bar{\theta }}}}(\delta ) \) shows that \(g'(\delta _{{\bar{\theta }}}) = 0 \), because \(f'_{{{\bar{\theta }}}}(\delta _{{\bar{\theta }}}) = 1\) and \(\delta _{{\bar{\theta }}} = f_{{\bar{\theta }}}(\delta _{{\bar{\theta }}})\). Moreover, we calculate \(g''(\delta ) = v_3''(\delta ) + v_3''(f_{{{\bar{\theta }}}}(\delta )) \, (f'_{{{\bar{\theta }}}}(\delta ))^2 + v_3'(f_{{{\bar{\theta }}}}(\delta )) \, f''_{{{\bar{\theta }}}}(\delta )>0\), where we used the monotonicity and strict convexity of \(v_3\) and the concavity of \(f_{{{\bar{\theta }}}}\), which follows from an elementary computation. This indeed implies (A.20).
This, along with (A.14), implies that (A.19) holds, with equality only if all bonds of an optimal cell have length \({\bar{\ell }}\), all angles have amplitude \({\bar{\theta }}\), and \(\delta _{13} = \delta _{24} = \delta _{{\bar{\theta }}}\). Clearly, for an optimal cell, \({\bar{\ell }}\) and \({\bar{\theta }}\) are given uniquely. We finally observe that \({\bar{\ell }} \le 1\) and \({\bar{\theta }} <\pi /2\). For \({\bar{\ell }}\), this follows from the fact that \(\ell ^* \le 1\), as shown in Step 4, and \({\bar{\theta }}<\pi /2\) has been checked in Step 5.
Step 8. Let us conclude by collecting some remarks on the function \(f(\varvec{\theta },\delta _{13})\) used throughout the proof. If \(\delta _{13} = \pi \), we have that \(\theta _1 + \theta _2 = \theta _3 + \theta _4 = \delta _{13} = \pi \). Therefore, \(\delta _{24}\) can be chosen arbitrarily in \([0,\pi ]\), see Fig. 14, and \(f(\cdot ,\pi )\) is hence not defined. For \(\delta _{13} < \pi \), the definition is given by
which can be derived by elementary, yet tedious, trigonometry. Let us now check that f is smooth for all \(\delta _{13} \in [\pi \lambda ,\delta ^*]\) and \(\varvec{\theta } \in [\pi /2\eta ,\pi ]^4\) such that \(f(\varvec{\theta } , \delta _{13}) = \delta _{24} \in [\pi \lambda ,\delta ^*]\) which we need in Step 6 of the proof. First, since \(\delta _{24} \in [\pi \lambda ,\delta ^*]\) the expression inside of \(\arccos \) is bounded away from \(1\) and 1. As \(\delta _{13} \in [\pi \lambda ,\delta ^*]\), \(\sin \delta _{13}\) is bounded away from 0. Thus, it suffices to check that the expressions inside the square roots are bounded away from 0. Indeed, \(\varvec{\theta } \in [\pi /2\eta ,\pi ]^4\) implies \(\cos \theta _1 \, \cos \theta _4 \rightarrow 0\) as \(\eta \rightarrow 0\) and \((\cos \theta _2  \cos \delta _{13} \cos \theta _1)(\cos \theta _3  \cos \delta _{13}\cos \theta _4)/\sin ^2\delta _{13} \ge 0\). As \(\cos (f(\varvec{\theta } , \delta _{13}))\) lies in a neighborhood of \(1\), this is indeed only possible if the value of each of the square roots is close to 1. \(\square \)
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Friedrich, M., Seitz, M. & Stefanelli, U. Tilings with Nonflat Squares: A Characterization. Milan J. Math. 90, 131–175 (2022). https://doi.org/10.1007/s00032022003505
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DOI: https://doi.org/10.1007/s00032022003505
Keywords
 Nonflat regular square
 Configurational energy
 Ground state
 Characterization
Mathematics Subject Classification
 92E10