TL;DR
This paper introduces a unified geometric approach using hyperbolic plane tessellations to classify various types of SL2-tilings and integer friezes, extending classical combinatorial models with new results.
Contribution
It develops a geometric framework based on Farey graph tessellations for classifying tame and positive SL2-tilings and integer friezes, providing new insights and models.
Findings
Classifies bi-infinite sequences as quiddity sequences of positive infinite friezes.
Provides a geometric analogue for classical combinatorial models of friezes.
Determines conditions for sequences to correspond to positive infinite friezes.
Abstract
Recently there has been significant progress in classifying integer friezes and -tilings. Typically, combinatorial methods are employed, involving triangulations of regions and inventive counting techniques. Here we develop a unified approach to such classifications using the tessellation of the hyperbolic plane by ideal triangles induced by the Farey graph. We demonstrate that the geometric, numeric and combinatorial properties of the Farey graph are perfectly suited to classifying tame -tilings, positive integer -tilings, and tame integer friezes -- both finite and infinite. In so doing, we obtain geometric analogues of certain known combinatorial models for tilings involving triangulations, and we prove several new results of a similar type too. For instance, we determine those bi-infinite sequences of positive integers that are the quiddity…
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Classifying -tilings
Ian Short
School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, United Kingdom.
2010 Mathematics Subject Classification: Primary 05E15; Secondary 11B57.
Key words: frieze, Farey graph, -tiling.
Abstract
Recently there has been significant progress in classifying integer friezes and -tilings. Typically, combinatorial methods are employed, involving triangulations of regions and inventive counting techniques. Here we develop a unified approach to such classifications using the tessellation of the hyperbolic plane by ideal triangles induced by the Farey graph. We demonstrate that the geometric, numeric and combinatorial properties of the Farey graph are perfectly suited to classifying tame -tilings, positive integer -tilings, and tame integer friezes – both finite and infinite. In so doing, we obtain geometric analogues of certain known combinatorial models for tilings involving triangulations, and we prove several new results of a similar type too. For instance, we determine those bi-infinite sequences of positive integers that are the quiddity sequence of some positive infinite frieze, and we give a simple combinatorial model for classifying tame integer friezes, which generalises the classical construction of Conway and Coxeter for positive integer friezes.
1 Introduction
The genesis of the subject of -tilings is Coxeter’s work on frieze patterns [Co1971]. To describe frieze patterns, we begin with Coxeter’s original motivating example, shown in Figure 1.1.
This is a bi-infinite strip of integers with bi-infinite rows of zeros at the top and bottom. Any four entries that form a diamond shape
[TABLE]
satisfy the unimodular rule . More generally, we define a positive frieze to be a finite list of bi-infinite sequences of integers – which we think of as the rows of an array such as Figure 1.1 – where all entries are positive except the first and last rows of zeros, and any diamond of four integers satisfies the unimodular rule. The order of the frieze is the number of rows minus one.
Coxeter proved in [Co1971] that every positive frieze is periodic, and later he and Conway [CoCo1973] classified positive friezes using triangulated polygons. A triangulated polygon is a convex Euclidean polygon partitioned into triangles by non-crossing diagonals; an example is shown in Figure 1.2.
The integers at the vertices of the polygon count the number of triangles in the partition that are incident to each vertex. Reading these integers in a clockwise fashion around the polygon, starting from the rightmost vertex, we obtain the sequence , which is the periodic part of the third row of Figure 1.1. Of course, we obtain a cyclic permutation of this triangle-counting sequence by starting from a different vertex. The quiddity cycle of a positive frieze is the cyclically-equivalent set of finite sequences that are the periodic part of the third row of the frieze. Conway and Coxeter proved that there is a one-to-one correspondence between triangulated -gons and positive friezes of order , in which the triangle-counting cycle of an -gon corresponds to the quiddity cycle of a positive frieze.
The subject of frieze patterns has grown enormously since Conway and Coxeter’s work, not least because of important connections between frieze patterns and cluster algebras; see, for example, [CaCh2006]. Assem, Reutenauer and Smith [AsReSm2010] introduced the concept of -tilings to this expanding field in their study of friezes and cluster algebras. The review article by Morier-Genoud [Mo2015] discusses the significance of frieze patterns in diverse disciplines.
Our starting point is the work of Morier-Genoud, Ovsienko and Tabachnikov [MoOvTa2015], who reproved Conway and Coxeter’s original result using the Farey graph and studied certain periodic -tilings. We take these geometric methods further, and offer a unified approach to classifying -tilings, bringing together recent work of Bergeron and Reutenauer on tame -tilings [BeRe2010], of Bessenrodt, Holm and Jørgensen [BeHoJo2017, HoJo2013] on positive integer -tilings, of Baur, Parsons and Tschabold [BaPaTs2016, BaPaTs2018, Ts2019] on infinite friezes, of Morier-Genoud, Ovsienko and Tabachnikov on antiperiodic tilings [MoOvTa2015], and of Ovsienko [Ov2018] on tame friezes with positive quiddity sequences. We demonstrate that the geometric, numeric and combinatorial properties of the Farey graph provide deep insight into -tilings.
This extended introduction describes the full suite of results for this classification programme, illustrating the theorems with examples and figures.
Fundamental to this approach is the special linear group of degree 2 over the integers, namely
[TABLE]
Informally speaking, an -tiling is a bi-infinite matrix such that any two-by-two submatrix belongs to . An example of an -tiling is shown in Figure 1.3.
Now we give a more precise definition of an -tiling, and to this end we consider functions , which represent bi-infinite matrices. It simplifies our presentation to write for the entry , for .
Definition**.**
An -tiling is a function such that
[TABLE]
for .
We think of as the entry of in the th row and th column, with the row index increasing downwards and the column index increasing from left to right – the usual conventions for matrices.
We restrict our attention to a class of -tilings called tame tilings, which possess rigidity properties essential to our approach.
Definition**.**
An -tiling is tame if
[TABLE]
for .
Our principal tool for classifying -tilings is the Farey graph. This is an infinite graph embedded in the extended complex plane . In future we write for (and use similar notation for and ). To describe the Farey graph precisely, we define to be the (open) upper half-plane, which is a model of the hyperbolic plane when it is endowed with the Riemannian metric (however, hyperbolic geometry will not feature explicitly in our arguments). Each hyperbolic line in this model of the hyperbolic plane is either the upper half of a circle centred on the real axis or a half-line in orthogonal to the real axis. Hyperbolic lines of the former type join one real number (on the boundary of ) to another, and hyperbolic lines of the latter type join a real number to .
We use the term reduced rational to describe an expression in which and are coprime integers. Each rational number can be expressed as a reduced rational in precisely two ways; for example, the rational number can also be written as . For convenience, we consider the expressions and to be reduced rationals, both representing the point .
The Farey graph is the graph with vertices and edges comprising those pairs and of reduced rationals for which . We represent the edge incident to and by the unique hyperbolic line in between those two boundary points. The collection of all edges creates a tessellation of by triangles, part of which is shown in Figure 1.4.
Next we discuss an action of on the Farey graph. Given and , we define
[TABLE]
with the usual conventions concerning . This formula specifies an action of on in the standard way. It also specifies an action of on the extended real line , on the extended rationals , and on the upper half-plane as a group of orientation-preserving hyperbolic isometries. The action is not faithful because and determine the same map; however, the action of the quotient group (the modular group) is faithful.
Suppose now that are integers that satisfy . Let
[TABLE]
Then
[TABLE]
so . In particular, if and are adjacent vertices of the Farey graph , and , then and are also adjacent vertices of . Consequently, we see that acts on too, and in fact each element of induces a graph automorphism of .
Given two vertices and of , we write if and are adjacent. A bi-infinite path in is a sequence of vertices of such that , for . We denote this path by . We also consider finite paths, defined in the obvious way, with similar notation. Each vertex can be written as a reduced rational in exactly two ways, and it is possible to choose such representations , for , such that , for . In fact, there are exactly two ways of doing this; the other way is , for .
In this paper we use paths in to classify -tilings. We now describe the details of this correspondence.
Evidently, if is an -tiling, then so too is the bi-infinite matrix , obtained by taking the negative of each of the entries of . For our purposes, it is helpful to identify these two tilings, as follows. We define to be the quotient set of the set of all tame -tilings by the equivalence relation that identifies and . And we write for an element of , where is understood to be a tame -tiling.
Next, consider the collection of all bi-infinite paths in . We will define a function from to . Consider a pair of bi-infinite paths . The vertices of are reduced rationals , for , and the vertices of are reduced rationals , for . We choose these representations such that and , for . Let us now define by
[TABLE]
where, as usual, . This is the fundamental formula underlying our approach; it is much the same as the formula used in [MoOvTa2015, Theorem 3], but without the modulus signs. One can check that is a tame -tiling (we do so in Section 2), and we define .
Notice that switching from to for all the vertices of changes the corresponding -tiling to . This justifies the need to identify in .
Let us consider an example, using the bi-infinite paths
[TABLE]
illustrated in Figure 1.5. The path has vertices , and has vertices , for , where , , and . The signs of these numbers have been chosen carefully to ensure that and , for . Then the corresponding matrix is that given in Figure 1.3, with entries
[TABLE]
for . In general, each [math] entry of corresponds to an intersection point of and ; in this particular case there is only one such point.
Since the image of a bi-infinite path under an element of is also a bi-infinite path, we see that acts on , and so it acts on too, by the formula , for and . Let be the quotient map.
In Section 2 we will verify that , for and . It follows that induces a map from to such that the following diagram commutes.
{\mathscr{P}\times\mathscr{P}}$${\textnormal{{SL}}_{\mathbf{2}}}$${(\mathscr{P}\times\mathscr{P})/\text{SL}_{2}(\mathbb{Z})}$$\pi$$\widetilde{\Phi}$$\Phi
This brings us to our first theorem.
Theorem 1.1**.**
The map is a one-to-one correspondence.
Let us briefly (and informally) discuss the inverse function of ; a more formal definition will follow later. Consider a tame -tiling . Choose any two consecutive rows of , and form a bi-infinite sequence of rational numbers by dividing each entry in the top row by the corresponding entry in the bottom row. This sequence is a bi-infinite path in the Farey graph , and another bi-infinite path can be obtained by choosing two columns instead of rows. After adjusting one of the two paths by applying a suitable element of , we obtain a pair of bi-infinite paths in that satisfy .
The characterisation of tame -tilings provided by Theorem 1.1 is comparable to that of [BeRe2010]*Proposition 3 (in two dimensions), but framed in the context of the Farey graph. The significance of Theorem 1.1 is that it gives us geometric insight into the structure of tame -tilings, and by restricting to subcollections of we can obtain precise descriptions of various classes of tilings.
For a first application of this procedure, we classify the set of positive -tilings (-tilings with positive integer entries), all of which are necessarily tame. A classification of positive -tilings has appeared before, using combinatorial techniques, in [BeHoJo2017]. Later (in Section 12) we will demonstrate that the geometric model of positive -tilings that we obtain using the Farey graph gives rise to the combinatorial model of [BeHoJo2017].
The set can be identified with a subset of in the obvious way (under which a positive -tiling is identified with ).
For the next definition we identify with the unit circle in the complex plane using the modified Cayley transform , which maps to , and sends to , [math] to , to , and to .
Definition**.**
A bi-infinite sequence of distinct points in is in clockwise order if the corresponding sequence of points in is such that is in clockwise order around , for all with .
A bi-infinite path in is a clockwise bi-infinite path if the sequence is in clockwise order.
The definition says, informally speaking, that a clockwise bi-infinite path traverses clockwise and does not complete more than one full cycle. We define clockwise order for finite and half-infinite sequences, and clockwise finite and half-infinite paths, in a similar way.
Each clockwise bi-infinite path converges (in ) in the backward direction to a point and in the forward direction to a point . These are called the backward limit and forward limit of , respectively, and they may be equal. Consider the collection of all pairs of clockwise bi-infinite paths for which is in clockwise order, where possibly and possibly (but and ). Let denote the collection of all such pairs. Observe that acts on .
By restricting the function to we obtain the following result.
Theorem 1.2**.**
The map is a one-to-one correspondence.
We demonstrate the correspondence of this theorem with two examples, illustrated in Figure 1.6. Each of the two subfigures displays a copy of the Farey graph mapped onto the unit disc by the modified Cayley transform , with vertex labels from . A pair of clockwise bi-infinite paths is highlighted in each Farey graph.
The paths shown in Figure 1.6(a) are
[TABLE]
The path has vertices and has vertices , for , where , , and . Here we have and , for . The limits of the two sequences are , , and , and the corresponding positive -tiling has entries
[TABLE]
for . This tiling is illustrated in Figure 1.7(7(a)).
The paths shown in Figure 1.6(b) are
[TABLE]
The path has vertices and has vertices , for , where , , and , so and . The limits of the two sequences are and , and the corresponding positive -tiling has entries
[TABLE]
for . This tiling is illustrated in Figure 1.7(7(b)).
To analyse these two tilings, we define a function by
[TABLE]
where and are reduced rationals. This function has been considered before in a related context, in [MoOvTa2015]. It is not a metric on . Evidently if and only if , and if and only if and are adjacent vertices of . Furthermore, if , where has vertices and has vertices , for , then we will see in Section 3 that is positive, so .
Now, in Figure 1.7(7(a)) the (unique) smallest entry is 2. Accordingly, 2 is the least value taken by for vertices of and of , where and are the paths shown in Figure 1.6(a). This value is achieved when and , and it is achieved at no other two vertices, one from and the other from (for a proof of a more general result of this type, see Theorem 12.3).
Consider now the paths and of Figure 1.6(b) and the corresponding tiling of Figure 1.7(7(b)). These two paths come within a -distance of from each other at the vertices [math] and , so the corresponding tiling has an entry 1. Again, this is the unique smallest entry, because the -distance between any other two vertices, one from and the other from , is greater than 1.
The paths and of Figure 1.6(b) are particularly special because they satisfy and . Moreover, both these points are rational numbers. We shall see later that in the special cases when or , the rationality or irrationality of each of these numbers is highly significant. For instance, if one of them is irrational, then there are infinitely many entries 1 in the corresponding -tiling (see Theorem 12.5). In Section 12 we discuss these properties further, and relate Theorem 1.2 to the combinatorial models used to classify positive -tilings in [BeHoJo2017, HoJo2013].
Let us move on to another example of application of Theorem 1.1, this time of infinite friezes, studied by Baur, Parsons and Tschabold in [BaPaTs2016, BaPaTs2018] and Tschabold in [Ts2019]. We encountered Coxeter’s frieze patterns earlier; infinite friezes are similar types of integer arrays, but instead of a border row of zeros at the bottom they continue downwards indefinitely. An example is shown in Figure 1.8.
There is a bi-infinite row of 0s followed by successive bi-infinite rows of integers, and each row is shifted relative to the rows above and below it in such a way that any four entries , , and in a diamond shape satisfy the unimodular rule , which we met earlier for Coxeter’s frieze patterns. For our purposes, it is convenient to consider infinite friezes as types of -tilings, and we can do this by, informally speaking, rotating an infinite frieze through clockwise to give ‘half’ the entries of a bi-infinite matrix (the entries , for ), with [math]s down the leading diagonal (, for ).
We now complete the other entries of by making it antisymmetric: , for . The resulting bi-infinite matrix is clearly an -tiling, and in fact if is tame, then it is the unique tame -tiling with entries , for .
This uniqueness property is proven in, for example, [BeRe2010, Proposition 21] (with the assumption, not needed here, that , for ); it also follows quickly from Theorem 1.1. Briefly, if is tame, then we can choose two paths with vertices and , for , where and , such that . The condition implies that either and for , or and for , from which it follows that , for .
Carrying out this process of completion for the infinite frieze in Figure 1.8 gives the -tiling displayed in Figure 1.9.
This discussion motivates the following definition, which uses slightly different terminology to that of [BaPaTs2016, BaPaTs2018, Ts2019] suitable for our purposes.
Definition**.**
An infinite frieze is an -tiling that satisfies , for . A positive infinite frieze is an infinite frieze for which , whenever .
Of principal interest to us are tame infinite friezes. These can be characterised as tame -tilings that satisfy , for , because, as we have seen, this condition implies the stronger antisymmetry condition , for .
It is sometimes convenient to consider an infinite frieze to be in the form of Figure 1.8, where the entries with are omitted; in this case we say that the frieze is in standard form. The entries of the second row of an infinite frieze in standard form are either all or all . It is the third row of the frieze that is of particular interest to us; if all entries of the second row are 1, then the third row is called the quiddity sequence of the infinite frieze.
Definition**.**
Suppose that is an infinite frieze with , for . The quiddity sequence of is the sequence , for .
If is an infinite frieze with , for , then the quiddity sequence of is defined to be the quiddity sequence of .
Let us examine the relationship between infinite friezes and pairs of bi-infinite paths in . Any infinite frieze satisfies , for . Using the formula , with the usual notation for bi-infinite paths, we have that , and hence , for . Thus, in this special case, the two bi-infinite paths coincide.
Now, the second row of (with entries , for ) consists entirely of s or entirely of s. Since , for , we see that the formula for gives rise to an infinite frieze whose second row comprises s, and the formula for gives rise to an infinite frieze whose second row comprises s. It does not matter which formula is used for computing images of , since switching from one to the other merely changes the sign of the corresponding -tiling. For consistency, we use the latter formula.
We define to be the collection of all tame infinite friezes, with, as usual, the friezes and identified. By identifying with a subcollection of using the map , we can think of as a function from to .
It is now a small matter to deduce the following theorem as a consequence of Theorem 1.1.
Theorem 1.3**.**
The map is a one-to-one correspondence.
Thus we see that while two bi-infinite paths are needed to specify a tame -tiling, only one bi-infinite path is needed to specify a tame infinite frieze. For example, consider the path
[TABLE]
illustrated in Figure 1.10. Let denote the th Fibonacci number, where , and , for . Let denote the bi-infinite sequence with equal to and for . Then has vertices , for , where and , and it can be verified that , for . The corresponding tame integer frieze has entries
[TABLE]
for . This is the infinite frieze shown in Figures 1.8 and 1.9.
The integer is marked just above the vertex [math] of the path in Figure 1.10 to indicate that, in navigating the Farey graph, takes the second right turn at [math]. And the integer is marked at the vertex because takes the first left turn at . Continuing in this way we can construct a bi-infinite sequence of integers, called the itinerary for , which comprises directions for to navigate the Farey graph, where we use positive integers for right turns and negative integers for left turns (and [math] for about turns).
We can formalise this definition as follows. Here we use the facts that the neighbours of in are the integers, and elements of are automorphisms of , so they preserve adjacency.
Definition**.**
Let be a bi-infinite path in . For each index , choose such that and , and define . The bi-infinite sequence of integers is called the itinerary for . The itinerary for a finite path is defined in a similar way.
We use square-bracket notation for itineraries because that notation is used in the theory of continued fractions, and itineraries are intimately related to integer continued fractions. Indeed, continued fractions featured significantly in Coxeter’s original work on positive friezes [Co1971], and they play an important role here too. For reasons of concision, however, we have elected to present our work without appealing to continued fractions. (See [MoOv2019] for a detailed account of the relationship between continued fractions, frieze patterns, triangulations and relations in the modular group.)
Observe that the itinerary of a bi-infinite path specifies the path uniquely, up to equivalence of paths.
The particular path illustrated in Figure 1.10 has itinerary , where is the entry of the sequence at index [math]. This sequence coincides with the quiddity sequence of the corresponding frieze (see the third row of Figure 1.8), and in fact this observation holds more generally (see Theorem 4.1). As a consequence, we deduce the relatively elementary fact that any bi-infinite sequence of integers is the quiddity sequence of some tame infinite frieze.
Next we consider positive infinite friezes (all of which are necessarily tame). Let be the set of clockwise bi-infinite paths, and let denote the collection of positive infinite friezes (which we identify with a subset of by the map ). Recall that the limits and of a path from may be equal. Given the results we have seen so far, the next theorem should come as no surprise.
Theorem 1.4**.**
The map is a one-to-one correspondence.
We demonstrate the correspondence of this theorem with two examples, illustrated in Figure 1.11. Each of the two subfigures shows a clockwise bi-infinite path in the Farey graph.
Figure 1.11(a) illustrates the path
[TABLE]
This path has vertices , where and , so , for , and the corresponding positive infinite frieze has entries
[TABLE]
for . It is displayed in Figure 1.12(a). Notice that the quiddity sequence of this frieze is , which is the itinerary for .
Figure 1.11(b) illustrates the clockwise bi-infinite path
[TABLE]
This path has vertices , for , where
[TABLE]
and it can be verified that , for . The corresponding positive infinite frieze with entries , for , is displayed in Figure 1.12(b). Again, the quiddity sequence of the frieze (namely ) is the itinerary for .
For the path in Figure 1.11(a), the two limits and are distinct. In contrast, for the path in Figure 1.11(b), and both equal , an irrational number (the reciprocal of the golden ratio). In Section 11, where we explore circumstances such as these more carefully, we shall see that in this case the entry 1 appears in infinitely many rows of the infinite frieze in standard form (see Theorem 11.3); in this particular case there is a 1 in every row.
Next we look at how to classify positive infinite friezes by their quiddity sequences. We recall from the start of this introduction that Conway and Coxeter used triangulated polygons to determine the (finite) quiddity cycles of positive friezes.
Definition**.**
A cycle sequence is a finite sequence of positive integers obtained by removing the final term from any one of the cyclically-equivalent sequences that makes up the quiddity cycle of some positive frieze. We say that a bi-infinite sequence of integers is acyclic if no cycle sequence appears as a finite subsequence of contiguous terms of .
For example, the quiddity cycle for the frieze of Figure 1.1 consists of and its cyclically-equivalent sequences. Thus is a cycle sequence, and so is . The phrase ‘cycle sequence’ is used because such sequences of positive integers correspond to itineraries of finite paths that are simple cycles in the Farey graph, as we shall see.
Theorem 1.5**.**
A bi-infinite sequence of positive integers is the quiddity sequence of a positive infinite frieze if and only if it is acyclic.
For an example, the bi-infinite sequence is not the quiddity sequence of a positive infinite frieze because it is not acyclic – it contains the cycle sequence as a subsequence of contiguous terms. On the other hand, because every cycle sequence has an entry 1 (which can be verified combinatorially using triangulated polygons), we see that any bi-infinite sequence of integers greater than 1 is acyclic and therefore is the quiddity sequence of some positive infinite frieze.
Another application of our procedure for classifying -tilings concerns tame friezes. A tame frieze is an integer frieze that is subject to a tameness condition, which we will come to shortly. And an integer frieze is simply a bi-infinite strip of integers of the same form as a positive frieze, but with integer entries, not necessarily positive. See Figure 1.13 for an example.
Just as for positive friezes, the order of an integer frieze is the number of rows minus one. Observe that the second and second-last rows are each composed entirely of 1s or entirely of s.
A feature of our approach is to consider friezes as types of -tilings; let us see how to do this for an integer frieze of order . Following the procedure for infinite friezes, we begin by rotating the frieze through clockwise to give a ‘diagonal strip’ of integers, which we will complete to a full bi-infinite matrix . Let us assume that the top row of [math]s of the integer frieze, once rotated, becomes the leading diagonal of terms , for , of , in which case the bottom row of [math]s becomes the diagonal of terms , for , and the other entries are , for .
Two possibilities arise. Either the entries from the second row of the integer frieze (all or all ) differ in sign from the entries in the second-last row, or else the entries in these two rows are all the same. In the first case, we complete by defining for , and in the second case we define for . Clearly, the resulting bi-infinite matrix is an -tiling. We will prove that if is tame then it is the unique completion of the integer frieze to a tame -tiling.
Suppose then that is any completion of the integer frieze , for , to a tame -tiling. Since , for , we see that is an infinite frieze, so there is a single bi-infinite path in with vertices (where ) such that , for . Since , for , it follows that , and hence , for . Consequently, either and for , or and for . Using the formula , we see that in the first case for , and in the second case for , so the completion of the integer frieze is indeed unique.
Figure 1.14 exhibits the completion of the integer frieze of Figure 1.13. The diagonals of 0s are shaded grey to highlight the strips of Figure 1.13 that make up the tiling.
We use the completed form of an integer frieze to define tame friezes more formally.
Definition**.**
Let be an integer greater than 1. A tame frieze of order is a tame -tiling such that , for .
A tame frieze is said to be in standard form if it is expressed as in Figure 1.13. With this representation, many of the entries of the frieze are omitted – they can be recovered using either the periodicity rules for , or the antiperiodicity rules for (whichever of the two sets of rules is appropriate for that particular frieze).
Notice that if is a tame frieze of order , then it is also a tame frieze of order , for any positive integer .
We consider now how to represent any tame frieze using the Farey graph. We have seen that any bi-infinite path in with vertices such that must satisfy , for . Hence is a periodic path with period (where is not necessarily the smallest period of the path). Conversely, it can easily be shown (and will be, later) that if is a periodic bi-infinite path with period , then the formulas , for , define a tame frieze of order .
We will relate periodic bi-infinite paths to closed paths.
Definition**.**
A closed path in is a finite path for which . A simple closed path in is a closed path in for which the vertices are distinct.
We define a clockwise simple closed path in to be a simple closed path in for which the sequence is in clockwise order.
Let denote the collection of closed paths of length in . We can identify with the collection of periodic bi-infinite paths of period in by mapping the closed path to the periodic bi-infinite path , where if .
We denote by the quotient set of the collection of tame friezes of order under the usual equivalence relation that identifies an -tiling with its negative. By thinking of as a collection of periodic bi-infinite paths, we can consider it to be a subcollection of (and hence a subcollection of using the identification ). Since acts on , we can consider to be a function from to .
Theorem 1.6**.**
The map is a one-to-one correspondence.
Positive friezes are special types of tame friezes, so we can restrict the function further to classify positive friezes. To do this, however, we first need to revisit the definition of a positive frieze in the context of -tilings. A positive frieze of order , then, is a tame frieze of order that satisfies , for . Notice that the entries of are not all positive; after all, , for , and, moreover, the antiperiodicity rules , for , imply that comprises alternating diagonal ‘strips’ of positive and negative integers separated by diagonals of [math]s.
Let denote the set of positive friezes, which we identify with a subset of using the embedding . We also define to be the collection of clockwise simple closed paths of length in , a subcollection of . The group acts on , so we can consider the restriction of to the set .
Theorem 1.7**.**
The map is a one-to-one correspondence.
Theorem 1.7 is equivalent to [MoOvTa2015, Proposition 2.2.1], which is itself a reformulation using the Farey graph of Conway and Coxeter’s original classification of positive friezes with triangulated polygons. In brief, any triangulated polygon can be realised as a hyperbolic triangulated polygon within , and the vertices of a hyperbolic triangulated polygon in listed in clockwise order specify a clockwise simple closed path in . We expand on this correspondence in Section 6; a full explanation is given in [MoOvTa2015]. The correspondence allows us to use the Farey graph and the class in place of the class of triangulated polygons.
Let us now work the other way and refashion Theorem 1.6 using triangulated polygons rather than closed paths in the Farey graph, thereby generalising Conway and Coxeter’s original classification of positive friezes to the larger class of tame friezes. Observe that any closed path in lies within some hyperbolic triangulated polygon in . Thus Theorem 1.6 shows that there is a correspondence between closed paths in triangulated polygons – where we allow ourselves the freedom to traverse the diagonals of a triangulated polygon – and tame friezes.
For example, consider the closed path illustrated in Figure 1.15(a). We can extend this path to a periodic bi-infinite path in the agreed way. The itinerary for is the periodic bi-infinite sequence with periodic part . This finite sequence of integers encodes a set of directions for navigating a closed path in a triangulated pentagon, as shown in Figure 1.15(b), where the labels are the terms of the itinerary for . The corresponding tame frieze is that displayed in Figure 1.13; observe that the quiddity sequence of the frieze and the itinerary for coincide.
In [CuHo2019, Section 7] an intriguing combinatorial model for classifying tame friezes is developed, which uses triangulations of polygons with integer labels attached to the triangles. It would be of interest to relate Theorem 1.6 to this model.
We finish this introduction with two other classes of -tilings that have been studied before and which can be considered within the geometric framework provided by the Farey graph. The first is a particular class of antiperiodic -tilings considered in [MoOvTa2015]. Let and be positive integers greater than 1. Let be an -tiling that satisfies the antiperiodicity rules , for , and which has an submatrix of positive integers; that is, there are integers and for which , for and . It is proved in [MoOvTa2015, Proposition 3.4.1] that all such -tilings are tame. We denote the class of these tilings under the usual equivalence relation that identifies and by . Let be the collection of pairs of clockwise simple closed paths in that do not intersect one another; this collection is invariant under the action of .
Theorem 1.8**.**
The map is a one-to-one correspondence.
Theorem 1.8 is equivalent to [MoOvTa2015, Theorem 2]; in the context of this paper, that theorem (effectively) prescribes two non-intersecting clockwise simple closed paths by their quiddity cycles.
A variety of other sets of periodic and antiperiodic -tilings can be classified in a similar spirit to Theorem 1.8. For example, the class corresponds to the set of tame -tilings that satisfy one of or for all , and one of or for all . Furthermore, one can refine this correspondence to distinguish periodic and antiperiodic rules, both horizontally and vertically (the method involves analysing clockwise simple closed subpaths of closed paths).
We pursue this no further, and instead turn our attention to another class of -tilings that has been considered before (in [Ov2018, MoOv2019]) – the class of those tame friezes of order for which the quiddity sequence comprises positive integers (but the other entries of the frieze are not necessarily positive). This class can be identified with a subclass of through the embedding , in which case we see that . The collection of closed paths corresponding to is the collection of closed paths of length for which is in clockwise order, for , where . (Such a path is not necessarily a ‘clockwise’ path, according to our definition earlier, since it might wind round the unit circle several times and thereby fail the condition that is in clockwise order whenever .) Observe that acts on , giving us the following result (due to the referee).
Theorem 1.9**.**
The map is a one-to-one correspondence.
An elegant combinatorial model for classifying was developed in [Ov2018], using dissections of polygons into -gons (triangles, hexagons, and so forth). In [MoOv2019, Sections 4 and 5] it is explained how this combinatorial model corresponds to the sort of model described by Theorem 1.9; indeed, [MoOv2019, Theorem 5.13] is almost equivalent to Theorem 1.9, although framed somewhat differently.
Acknowledgements
I gratefully acknowledge Peter Jørgensen for introducing me to friezes and -tilings, and for discussions in which many of the ideas presented here were conceived. I also thank the referee for recommending the inclusion of Theorems 1.7–1.9, as well as several other valuable contributions.
2 Tame tilings
Here we prove Theorem 1.1; that is, we prove that the function is a bijection. Some of our arguments could be shortened by extracting results from the literature. However, we have elected not to use other sources in order to keep the discussion self-contained.
We begin by stating (but not proving) a well-known elementary lemma that characterises tame -tilings.
Lemma 2.1**.**
An -tiling is tame if and only if there are bi-infinite sequences of integers and such that
[TABLE]
for .
Informally speaking, this lemma states that, for the matrix , ‘row plus row equals times row ’ and ‘column plus column equals times column ’.
The next lemma is also elementary.
Lemma 2.2**.**
Let , and be reduced rationals such that and . Then there is an integer such that and .
Proof.
The result can easily be established if or , so let us suppose that neither nor is 0. Combining the equations and we obtain . Since and are coprime we see that , for some integer . Consequently , as required. ∎
We will find it convenient to define
[TABLE]
We write for the transpose matrix of . Then clearly . Also, we often have recourse to the useful formula , for .
Let us now prove that the function is well-defined. Consider a pair of bi-infinite paths in the Farey graph . We continue the notation of the introduction, writing the vertices of as reduced rationals , for , and writing the vertices of as reduced rationals , for . We choose these representations such that and , for . Now we define by
[TABLE]
where . Then
[TABLE]
Hence is an -tiling. Next, observe that, by Lemma 2.2, for each there is an integer such that and . Hence
[TABLE]
for . Similarly, there are integers with , for . Hence is tame, by Lemma 2.1.
Notice that if we write the vertices of in the alternative form , for , then we obtain the matrix in place of (and a similar comment applies to ), so is indeed well-defined.
The next task is to check that two pairs of bi-infinite paths that lie in the same orbit of are mapped to the same element of . To see this, suppose that , and
[TABLE]
for . Then we can use the formula to give
[TABLE]
as required. It follows, then, that we have a well-defined function , which maps the orbit of under to .
For the second part of the proof of Theorem 1.1, we will define a function
[TABLE]
that will prove to be the inverse function of . The next lemma is fundamental to defining .
Lemma 2.3**.**
Let be a tame -tiling with
[TABLE]
Define , , and , for . Then
[TABLE]
for .
Proof.
Let be the bi-infinite sequence from Lemma 2.1. Then
[TABLE]
for . We will prove the lemma when (the case can be handled similarly). For ,
[TABLE]
Applying this with and we obtain
[TABLE]
Hence
[TABLE]
Taking the first component of each side of this vector equation gives the result. ∎
Consider now a tame -tiling . Let
[TABLE]
We define two bi-infinite sequences of reduced rationals and , for , by
[TABLE]
Then, as vertices of the Farey graph, for , since . Hence , for , determines a bi-infinite path in the Farey graph, as does , for (for similar reasons). Clearly, gives rise to the same pair of bi-infinite paths, so we obtain a function from to . We define to be the composition of the function just constructed with the projection .
It remains to prove that and are inverse functions. Consider, then, a tame -tiling . Under , the pair maps to the -orbit of the pair of bi-infinite paths with vertices and given by equations (2.2). And under this orbit maps to , where
[TABLE]
Now, , so we see that
[TABLE]
by Lemma 2.3. Hence is the identity function.
Next, consider a pair of bi-infinite paths and in with vertices and , respectively, where and , for . Let be the tame -tiling given by , for . Under , the -orbit of maps to the -orbit of , where and have vertices and , respectively, that satisfy
[TABLE]
and
[TABLE]
for . Now, applying equation (2.1) with gives
[TABLE]
Using this equation, and the formula , for , one can check that
[TABLE]
Let be the element of given by either side of this equation. Then
[TABLE]
for . It follows that and belong to the same orbit of , so is the identity function.
In summary, and are inverse functions, so is a one-to-one correspondence, which proves Theorem 1.1.
3 Positive tilings
In this section we prove Theorem 1.2, which says that is a one-to-one correspondence. To establish this, we first show that maps into , and then we show that maps into .
First, however, it is helpful to note that a clockwise bi-infinite path in has one of three types: (i) a decreasing bi-infinite sequence of rational numbers; (ii) , for some integer ; or (iii) for some integer and . Of course, in all three cases and are adjacent in , for .
Returning to the proof of Theorem 1.2, consider a pair of clockwise bi-infinite paths in the Farey graph, where, in the notation of the introduction, the limits are in clockwise order in (after identifying with the unit circle by the modified Cayley transform ) and and . By applying an element of we can assume that the vertex of with index 1 is . It follows that is a decreasing sequence of reduced rationals , where . Let us choose all the denominators to be positive, in which case the fact that the path is decreasing implies that , for . Consequently, we see that , for .
Let , for , be reduced rationals that form the path , where and . We choose the denominators to be positive if , and negative if . Again, we can check that , for . Now, because , for and , we obtain
[TABLE]
Similarly, if . In this way we see that maps into .
It remains to prove that maps into . Suppose then that is a positive -tiling. We will need the inequalities
[TABLE]
for , which follow immediately from the equations and the fact that the entries of are positive.
Observe that the image of under is the -orbit of the pair of bi-infinite paths and with vertices and , respectively, given by equations (2.2). By the left-hand inequality of (3.1), the first path is a decreasing sequence of positive rational numbers, so it is a clockwise bi-infinite path in . For the second path , we have
[TABLE]
for . In particular, and , and and . Observe that
[TABLE]
Using the right-hand inequality of (3.1), we see that for , and for . From the equations , for , we can infer that the sequence is increasing and the sequence is decreasing.
Next, since , for , we see that for , and for . It follows that the limits , , and of and are real numbers that satisfy . Therefore , like , is a clockwise bi-infinite path in , and the -orbit of lies in , as required.
This completes the proof of Theorem 1.2.
4 Tame infinite friezes
In this section we prove Theorem 1.3, which says that is a bijective function. This map sends the -orbit of a bi-infinite path (with vertices such that ) to , where is the tame -tiling given by
[TABLE]
for . Since , for , we see that is a tame infinite frieze.
Conversely, if is a tame infinite frieze, then it is a tame -tiling, so we can find two bi-infinite paths and in for which, with the usual notation,
[TABLE]
for . Since , for , it follows that , so . Therefore the preimage of under is an element of . This completes the proof of Theorem 1.3.
We conclude this section with the following theorem, referred to already in the introduction.
Theorem 4.1**.**
Let be a bi-infinite path in and let be a tame infinite frieze such that . Then the itinerary for is equal to the quiddity sequence of and .
Proof.
Let with itinerary . As usual, we write as a reduced rational , for , where .
Consider some particular integer . By applying an element of to (which preserves its itinerary) we can assume that and , in which case . In these circumstances either
[TABLE]
or else the sign of each of these numbers is reversed. In both cases we have
[TABLE]
Thus the th entry of the quiddity sequence , for , is equal to the th entry of the itinerary for . It follows that the quiddity sequence and itinerary coincide. ∎
5 Positive infinite friezes
In this section we prove Theorem 1.4, which says that the map is a one-to-one correspondence.
Consider a clockwise bi-infinite path , with vertices given by reduced rationals , for . By applying an element of , we can assume that the vertex of with index 1 is , and and . Let us choose the denominators to be positive if , and negative if . Then we can deduce that , for .
Suppose now that with . If and are both greater than 1 or both less than 1, then , so the tame infinite frieze corresponding to satisfies
[TABLE]
Similarly it can be shown that if , or if and one of or equals 1. Hence is a positive infinite frieze.
Conversely, suppose that is a positive infinite frieze, and let be the bi-infinite path in given by equation (2.2) with vertices , where and , for (which is in the preimage of under ). Observe that , , and . Also, and are negative for and they are positive for . For we have
[TABLE]
so is a decreasing sequence. Similarly, it can be shown that is an increasing sequence. Furthermore, if and then is negative, so
[TABLE]
Hence . It follows that is a clockwise bi-infinite path.
This completes the proof of Theorem 1.4.
6 Quiddity sequences of positive infinite friezes
Here we prove Theorem 1.5, which characterises the quiddity sequences of positive infinite friezes. In the course of the proof, and elsewhere in this work, we often use the following elementary lemma, the proof of which is omitted.
Lemma 6.1**.**
Let and be adjacent vertices of . Then any finite path whose initial and final vertices lie in different components of must pass through one or both of and .
Next we define hyperbolic triangulated polygons, which are hyperbolic versions of Euclidean triangulated polygons.
Definition**.**
A hyperbolic triangulated polygon comprises the sets of vertices and edges obtained from the union of finitely many triangles in the Farey graph, where any two of these triangles can be connected to one another by a finite sequence of successively adjacent triangles from the union.
There is another way to think about hyperbolic triangulated polygons, using simple closed paths in . Recall that a simple closed path in is a path in such that and are distinct. The length of this path is . It can easily be shown that the vertices of a hyperbolic triangulated polygon, listed in clockwise order, determine a clockwise simple closed path of length at least three (see [MoOvTa2015, Section 2.2] for an explanation). Conversely, any clockwise simple closed path of length at least three determines a hyperbolic triangulated polygon with vertices given by the vertices of the path, and the edges of the triangulated polygon are those edges of that connect pairs of vertices from the path.
There is a correspondence between hyperbolic and Euclidean triangulated polygons, which is illustrated in Figure 6.1 and can be described (in brief) as follows. To obtain a Euclidean triangulated polygon from a hyperbolic triangulated polygon we map the Poincaré disc model of the hyperbolic plane (which we are using) to the Klein disc model of the hyperbolic plane in the standard way (by projecting stereographically onto a hemisphere above the disc and then projecting orthogonally back onto the disc). This projection preserves the unit circle pointwise and it sends lines in the Poincaré model to straight Euclidean lines in the Klein model. Thus is taken to a Euclidean triangulated polygon with the same associated triangle-counting cycle.
Reversing this procedure, given a Euclidean triangulated polygon, one can obtain a hyperbolic triangulated polygon with the same triangle-counting sequence by an inductive process, starting from a single triangle and then gluing further triangles one at time. This process is formalised in [MoOv2019] (in particular, Section 2.4 of that paper), where an algorithm using Farey sums is used to achieve this purpose.
In this manner we see that for each Euclidean triangulated polygon there is a hyperbolic triangulated polygon of the same combinatorial type, and vice versa (this was observed in [MoOvTa2015]). From now on we use hyperbolic triangulated polygons rather than Euclidean triangulated polygons.
Recall from the introduction that the itinerary of a finite path is the sequence , where, for each index , the integer is the image of under a map that satisfies and . Also, recall that a cycle sequence is obtained by removing the final term from a quiddity sequence of some positive frieze.
Lemma 6.2**.**
A finite path in is a clockwise simple closed path if and only if the itinerary of is a cycle sequence.
Proof.
Let , with itinerary .
Suppose first that is a clockwise simple closed path. Let be the hyperbolic triangulated polygon with vertices those of . Let be the number of triangles in incident to . Then is a cycle sequence.
Consider any integer between (and including) 1 and . By applying an element of , we can assume that , and . The sequence is in clockwise order, so is a positive integer, and the remaining vertices of lie between [math] and . By applying Lemma 6.1 to the path with and equal to each of the integers in turn, we see that must pass through all the vertices . It follows that there are precisely triangles in that are incident to . Hence , and therefore the itinerary of is a cycle sequence.
Conversely, suppose that is a cycle sequence. Then there is some hyperbolic triangulated polygon with vertices such that (where ) is a clockwise simple closed path and such that there are triangles in incident to , for .
By applying an element of to we can assume that and . Now choose such that and . Then , by definition of , but also , because there are triangles in incident to . Hence . Repeating this argument we see that and are equal, so is a clockwise simple closed path. ∎
We can now prove Theorem 1.5, which says that a bi-infinite sequence of positive integers is the quiddity sequence of a positive infinite frieze if and only if it is acyclic.
Choose any bi-infinite sequence of positive integers and let
[TABLE]
be a bi-infinite path in with itinerary . For each , let denote the open interval of those points for which are in clockwise order in . Since for we can see (by mapping to [math], to , and to ) that and are disjoint. Two possibilities arise: either all the intervals are disjoint, or they are not. Let us consider each case in turn.
If the intervals are all disjoint, then is a clockwise bi-infinite path (as defined in the introduction). It follows that does not contain a clockwise simple closed path as a subpath of contiguous vertices of , so the itinerary of does not contain a cycle sequence as a subsequence of contiguous terms, by Lemma 6.2. Therefore the quiddity sequence of the corresponding frieze is acyclic.
Suppose now that the intervals are not disjoint. Then there are indices and with for which is either equal to , or else it lies in . However, by Lemma 6.1, the path cannot enter without first passing through or . Thus if , then there is some index with for which is equal to one of or . Thus we obtain a clockwise simple closed path as a subpath of contiguous vertices of . We can now apply Lemma 6.2 to see that the itinerary of contains a cycle sequence, so it is not acyclic.
This completes the proof of Theorem 1.5.
7 Tame friezes
Here we prove that is a bijective map from to (Theorem 1.6).
Suppose that is a closed path in (so ). As stated in the introduction, we identify this closed path with the periodic bi-infinite path , where if . Using the usual notation, we write with , for . For each integer , we have , so , and hence . Since also, we see that is a tame frieze of order .
Conversely, given a tame frieze of order we have already seen in the introduction that the preimage of under is the -orbit of a periodic bi-infinite path of period . This completes the proof of Theorem 1.6.
8 Positive friezes
We prove here that is a one-to-one correspondence (Theorem 1.7).
To this end, suppose that is a clockwise simple closed path in , and consider the periodic bi-infinite path defined by the property that if . As ever, we write as a reduced rational , for , where , and we let be the tame frieze of order with entries , for .
Choose two particular integers and with . By applying an element of , we can assume that and (and ). Then . Since is a clockwise simple closed path of length , we see that . A simple inductive argument shows that are all positive, so .
In summary, is a tame frieze of order that satisfies , for ; hence is a positive frieze of order .
Conversely, suppose that is a positive frieze of order . By Theorem 1.6 we can find a closed path , where and , for , with , for , such that , for . Furthermore, by applying an element of we can assume that and and . Observe that, for ,
[TABLE]
Hence
[TABLE]
for . That is, , so is a clockwise simple closed path, as required. This completes the proof of Theorem 1.7.
9 Antiperiodic tilings
This section has a proof that is a bijective map from to (Theorem 1.8).
Before proving this theorem we make a simple observation. Let be the shift map acting on the collection of bi-infinite paths in the Farey graph; sends the path with vertices to the path with vertices . Note that is a bijective map, so, for integers and , we can define an action of on in the obvious way.
Next, we define to be the self-map of that sends to , where , for . It is easily verified that
[TABLE]
Since is invariant under and is invariant under , we can use the displayed equation above to simplify our proof of Theorem 1.8.
Let us now prove that theorem. Suppose then that ; that is, suppose that , , and and do not intersect. We write , where is a clockwise simple closed path and for , and , where is a clockwise simple closed path and for . We use the usual notation and . Since and do not intersect, we can apply Lemma 6.1 to see that there are two consecutive vertices and of and two consecutive vertices and of such that is contained in one of the components of and is contained in one of the components of . Then, by applying to , we can assume that .
Next, by applying an element of , we can assume that and . Then is a decreasing sequence of negative numbers, and of course . Recall that there are two possible choices for the integer sequences and that differ only in sign. We choose the sequence for which (and ). That is a decreasing sequence, together with the equations , then implies that are all positive. Furthermore, we can see that and , for (the sign changes each cycle).
Similarly, it can be shown that is a decreasing sequence of positive numbers, that are positive numbers, and that and , for .
Let be the tame -tiling with entries , for . Then , and likewise , for . Furthermore, for and , we have , so
[TABLE]
Also, . Hence , for and , so .
Conversely, consider a tame -tiling . Then satisfies the antiperiodicity rules , for . Also, by applying a suitable shift map to , we can assume that , for and . Let and be the bi-infinite paths in defined by equations (2.2), with the usual notation for paths. Then and , for , so is periodic with period . Similarly, is periodic with period . Observe that and do not intersect, because has no entries [math].
Now, , for , and and are positive, so, after dividing throughout by we see that
[TABLE]
We know that and hence is a clockwise simple closed path. In a similar manner it can be shown that is a clockwise simple closed path. Hence .
This completes the proof of Theorem 1.8.
10 Tame friezes with positive quiddity sequences
In this section we prove Theorem 1.9, which says that is a bijective map from to . We use the following lemma.
Lemma 10.1**.**
Let be a bi-infinite path, with and , for , and let be the corresponding infinite frieze with entries , for . For each integer , the sequence is in clockwise order if and only if .
Proof.
Choose any integer . By applying an element of to , which will not affect , we can specify that . Let us also fix signs such that and . Then and . Hence and . The sequence is in clockwise order if and only if ; that is, if and only if . Since
[TABLE]
we see that is in clockwise order if and only if . ∎
Let us now prove Theorem 1.9. Applying Theorem 1.6, we choose a periodic bi-infinite path of period and a corresponding tame frieze of order with entries . By Lemma 10.1, the sequence is in clockwise order for all if and only if for all . That is, if and only if . This completes the proof of Theorem 1.9.
11 Combinatorics of positive infinite friezes
In this section we resume the discussion of the introduction relating our geometric characterisation of positive infinite friezes using the Farey graph to the combinatorial model of such friezes from [BaPaTs2016, BaPaTs2018]. The class of combinatorial objects that we need are triangulations of infinitely-many-sided polygons – or apeirogons, as they are often called.
Definition**.**
A hyperbolic triangulated apeirogon comprises the sets of vertices and edges obtained from the union of infinitely many triangles in the Farey graph, where any two of these triangles can be connected to one another by a finite sequence of successively adjacent triangles from the union.
We usually omit the adjective ‘hyperbolic’ and simply write ‘triangulated apeirogon’.
To appreciate the structure of triangulated apeirogons, it helps to consider the dual graph of the Farey graph, which is an infinite trivalent tree. Each vertex of this tree corresponds to a triangle of the Farey graph, and a triangulated apeirogon corresponds to an infinite connected subtree of the dual graph.
Let us explore a method for constructing a triangulated apeirogon from a clockwise bi-infinite path . For each integer , the open interval in that runs clockwise from to contains and all but finitely many neighbours of in the Farey graph . The set of vertices of consists of all the vertices , and, for each integer , all the neighbours of in that lie outside (some of which may well be other vertices of ). The edges of are those edges of that connect the vertices of . See Figures 11.1 and 11.2 for illustrations of this construction.
It follows quickly from the definition that is a triangulated apeirogon. The vertices of occur in clockwise order around , with no other vertices of in between. The remaining vertices of themselves form a path, as Theorem 11.2, to follow shortly, confirms. Before stating this theorem, we introduce an elementary lemma.
Lemma 11.1**.**
Given any irrational number there is an increasing sequence of rationals with limit and a decreasing sequence of rationals with limit such that and are adjacent in , for each .
This lemma could be proved, for example, by choosing and to be the odd and even subsequences of the sequence of convergents of the regular continued fraction expansion of . We omit the details.
In the next theorem we describe a clockwise path as forward-finite if it has a final vertex and forward-infinite if it has no final vertex and so continues indefinitely in the forward direction.
Theorem 11.2**.**
Let be a clockwise bi-infinite path and let be the associated triangulated apeirogon. Those vertices of that are not in form a clockwise path .
- (i)
Suppose that . If is rational, then comprises the single vertex , and if is irrational then is the empty set. 2. (ii)
Suppose that . If is rational, then is forward-finite with final vertex , and if is irrational then is forward-infinite with forward limit .
An analogous statement holds for the backward end of .
Proof.
Let , and let denote the set of those vertices of that are not vertices of .
Suppose first that . Since there are no vertices of in-between any two vertices and of (in a clockwise sense), we see that comprises at most one point, itself. If is irrational then it cannot be a vertex of . If is rational, then, after applying an element of we can assume that . In this case we see from Lemma 6.1 (with and ) that must pass through 0, so , because is adjacent to [math] in . This proves statement (i).
Suppose now that . The set lies within the closed interval that runs clockwise from to . An argument similar to that just applied shows that if (and only if) it is rational. Next, the set cannot accumulate at any points of other than and . This follows from the fact that accumulates only at and , and, in the disc model of the Farey graph, there are only finitely many edges of the Farey graph of Euclidean diameter greater than any given positive number. As a consequence, we can write the elements of as , where this sequence is in clockwise order and it may be finite, half-infinite or bi-infinite.
Consider now two consecutive terms and ; we will prove that they are adjacent in . Let be the largest index such that is adjacent to in . After applying an element of we can assume that and ; in which case, by the clockwise ordering, is equal to , for some positive integer , and is a negative integer. Suppose that . Then, by Lemma 6.1 (with and ), we must have for some . But is adjacent to , so this contradicts the definition of . Hence . Now, since [math] is a vertex of , and is adjacent to [math], we see that is a vertex of . On the other hand, none of the integers are adjacent to any vertex of , so they are not vertices of . Hence , and and are indeed adjacent in . It follows that is a clockwise path in .
If is rational, then (as we have seen) it is a vertex of , and it must be the final vertex, since is a clockwise path. If is equal to some irrational , then we can choose increasing and decreasing sequences of rationals and both with limit such that and are adjacent in for each index . By Lemma 6.1, the path must pass through one of or for all but finitely many indices . In fact, since we see that must pass through (for large ). Hence is a vertex of , because it is adjacent to in . In this way we see that is forward-infinite, with forward limit . This proves statement (ii). ∎
The path in Theorem 11.2 is in a sense dual to . This concept of a dual path in this context could be formalised, but we will not do so here.
We illustrate Theorem 11.2 with two examples, shown in Figure 11.1. The first of these two figures is repeated from Figure 1.11(a).
Figure 11.1(a) illustrates the path
[TABLE]
with vertices , where and , for . For this path, the limits and are both rational. The triangulated apeirogon constructed from is shown in Figure 11.2(a). The dual path is a finite path.
Figure 11.1(b) illustrates the path
[TABLE]
with vertices , where and , for (here is if , 0 if , and if , and is the th Fibonacci number). In this case the limits and are both irrational. The triangulated apeirogon constructed from is shown in Figure 11.2(b). The dual path is a bi-infinite path.
Triangulations of essentially the same type as those considered here appear in [BaPaTs2016, BaPaTs2018], where they are defined combinatorially; however, in that work the underlying geometrical object is an infinite strip rather than a disc (or half-plane). This is not a significant difference because any triangulated apeirogon can be transported between a disc and a strip using a conformal map. Indeed, an infinite strip is a particularly suitable space for a triangulation constructed from a clockwise bi-infinite path when the limits and are distinct, because those limit points can then be chosen to correspond to the two infinite ends of the strips. And in that case, the vertices of and occupy opposite sides of the strip boundary. From our geometric perspective, the strip model is less desirable when the backward and forward limits coincide.
It was observed in [BaPaTs2016]*Theorem 5.2 that the quiddity sequence of a positive infinite frieze can be read off from the corresponding triangulation. We can explain this in our terminology as follows.
Consider a triangulated apeirogon constructed from a clockwise bi-infinite path . Let be the positive infinite frieze corresponding to . By Theorem 4.1, the itinerary for is equal to the quiddity sequence of . But the th term of the itinerary just records the number of triangles in that are incident to the th vertex of . Thus we can obtain the quiddity sequence from an infinite form of the triangle-counting procedure originally employed by Conway and Coxeter.
We finish this section with a theorem that characterises clockwise bi-infinite paths for which and are equal and irrational by using the corresponding positive infinite frieze . One can obtain similar results to classify the other types of paths that we have considered here (with rational/irrational limits which may or may not coincide) using the corresponding friezes, but to go into this now would draw out this discussion beyond reasonable bounds.
Theorem 11.3**.**
Let be a clockwise bi-infinite path and let be the positive infinite frieze such that . The following statements are equivalent:
- (i)
* and coincide and are equal to an irrational number* 2. (ii)
there are increasing sequences of positive integers and with , for .
Proof.
We use the usual notation , with , for , where .
Suppose first that and coincide and are equal to an irrational number . By Lemma 11.1 we can choose increasing and decreasing sequences of rationals and both with limit such that and are adjacent in for each index . Now, the sequence is eventually increasing, with limit , and the sequence is eventually decreasing, again with limit . By Lemma 6.1, both sequences must pass through one of the points and , for all but finitely many indices , and, evidently, because of the ordering, the first sequence passes through and the second through . Given any sufficiently large positive integer , then, there are positive integers and for which and , so and are adjacent and
[TABLE]
Since this is so for all but finitely many pairs and , we see that there are infinitely many positive integers and (all distinct) with .
For the converse, suppose there are increasing sequences of positive integers and with , for . It follows that is adjacent to , for each . Now, for any positive constant there are only finitely many edges of the disc model of the Farey graph of Euclidean diameter greater than , so we see that the convergent sequences and converge to the same limit . Hence .
Suppose . By applying an element of , we can assume that . Then is a decreasing sequence, unbounded below, and is an increasing sequence, unbounded above. However, this is impossible, because cannot be adjacent to for all under these circumstances. Therefore is irrational. ∎
For an example of Theorem 11.3 in action, refer back to the path illustrated in Figure 1.11(b) and the corresponding positive infinite frieze of Figure 1.12(b). This frieze satisfies
[TABLE]
for , by Cassini’s identity for Fibonacci numbers, so there is a 1 in every row of the frieze in standard form.
12 Combinatorics of positive -tilings
A widely-used tool in the theory of -tilings is Conway–Coxeter counting, which, for a hyperbolic triangulated apeirogon , can be described as follows. Choose any vertex of . We assign the value [math] to that vertex. To each neighbour of in we assign the value . Next we carry out the following recursive process to assign a value to any particular vertex of . If is adjacent to two vertices of that have already been assigned the values and , then we assign the value to . We denote the value at from this counting procedure by .
Every vertex of can be assigned a value that is uniquely specified by this process. We do not prove this fact, although it is not difficult. Actually, it can be established using similar ideas to those used to prove the next lemma. This lemma demonstrates how the arithmetic of the Farey graph neatly encapsulates Conway–Coxeter counting. We recall from the introduction that for reduced rationals and , .
Lemma 12.1**.**
Let and be vertices of a triangulated apeirogon. Then .
Proof.
Observe first that both and are invariant under , in the sense that
[TABLE]
for , where and are vertices of the triangulated apeirogon . Hence we can assume that .
Let us write as a reduced rational with . Then . We will prove that also.
All vertices adjacent to have the form , for . Reduced rationals for other vertices in the Farey graph can be obtained by Farey addition: if and are neighbours in , then the unique vertex of between and on the real line for which , and form a triangle in is . It is well known (and reasonably straightforward to prove) that all vertices of the Farey graph can be obtained in this way. Conway–Coxeter counting simply records the denominators of this repeated process of Farey addition, so is equal to the denominator of , namely . ∎
In the previous section we described how to construct a triangulated apeirogon from a clockwise bi-infinite path. Now we discuss how to obtain a triangulated apeirogon from an element of – a pair of clockwise bi-infinite paths. First we carry out the procedure from the previous section to give two triangulated apeirogons and , the first for and the second for . It will help to think of these apeirogons as infinite connected subtrees of the dual graph of the Farey graph. Consider the unique minimal path in the dual graph connecting to (which is empty if and share a common triangle). We define a new triangulated apeirogon to consist of the vertices and edges from the collection of all triangles in , and this minimal path. One can verify that is indeed a triangulated apeirogon, and none of the vertices of lie in-between any two consecutive vertices of or any two consecutive vertices of (in the clockwise sense).
Using a result similar to Theorem 11.2 it can be shown that the vertices of the triangulated apeirogon can be split into four clockwise paths: and , and two other paths and , each of which may be empty, finite, half-infinite or bi-infinite. The paths themselves occur in the order clockwise in , and the only accumulation points of vertices of are the four limit points , , and . Precisely which configuration occurs depends on which if any of the four limit points are equal, and whether each one is rational or irrational.
There are many cases, and rather than discussing them all in detail, we supply a single illustrative example. (See [BeHoJo2017] for a more-detailed look at the combinatorial configurations, from a different perspective.) Consider the paths
[TABLE]
in Figure 12.1(a). Here and are both rational, and and are both irrational. Figure 12.1(b) shows the corresponding triangulated apeirogon . Since is irrational and is rational, the vertices of in between and form a half-infinite path with backward limit and (forward) final vertex . Likewise, the vertices of in-between and form a half-infinite path with initial vertex and forward limit .
These triangulated apeirogons constructed from pairs of clockwise bi-infinite paths are of the same type as the triangulations considered by Bessenrodt, Holm and Jørgensen in [BeHoJo2017]. Those authors proved that every positive -tiling can be obtained by choosing such a triangulation and applying Conway–Coxeter counting. Using Lemma 12.1 and Theorem 1.2 we can see (in essence, if not with full rigour) that the method for classifying positive -tilings using the Farey graph is equivalent to that of [BeHoJo2017]; the geometry and numerics of the Farey graph provide a short cut for simplifying combinatorial arguments.
It was observed in [BeHoJo2017] that any positive -tiling without any entries equal to has a unique smallest entry. We reprove that result here using the Farey graph. The following lemma is at the heart of the proof.
Lemma 12.2**.**
Let and be adjacent vertices of , and let and be vertices that lie in distinct components of . Then .
Proof.
By applying an element of we can assume that . Then and are rationals, and let us suppose that , so lies inside the interval (the argument for is much the same). We remarked earlier that every rational inside can be obtained by repeatedly applying Farey addition, starting from the pair and . Observe that the Farey sum of two rationals with positive denominators is a rational with a larger denominator. Thus, writing and (in reduced form, with ) we see that . The result follows, since and . ∎
Theorem 12.3**.**
Let be a positive -tiling without any entries equal to 1. Then has a unique smallest entry.
Proof.
By shifting the indices of entries of we can assume that assumes its least value at . We use the usual notation for paths and with . By applying an element of we can assume that the zeroth vertex of is .
Let be the zeroth vertex of . Since , it follows that and are not adjacent in , so is not an integer. Hence has precisely two neighbours and in with denominators of smaller magnitude than . These two rationals are called the Farey parents of in some sources, such as [BeHoSh2012]. They lie on either side of – let us say that – and they are themselves neighbours in .
Neither of the paths nor can pass through or , because if (say) did then there would be a vertex with , by Lemma 12.2. It follows from Lemma 6.1 that lies within the interval and lies in the complement in of this interval.
Consider now any vertex of other than and any vertex of . Then lies in either or . In both cases we can apply Lemma 12.2 to see that . This shows that the minimum value of is only achieved when . Applying the same argument with the roles of and reversed shows that the minimum is only achieved at , as required. ∎
In contrast to Theorem 12.3, a positive -tiling can have infinitely many entries 1. We characterise the -tilings of that type using the Farey graph in the next two results.
Lemma 12.4**.**
Let be a positive -tiling, and suppose that for some integers and . If and , or and , then .
Proof.
Let be a pair of bi-infinite clockwise paths in (with the usual notation) such that . Then and are adjacent in . Suppose that and . Observe that , , , are in clockwise order in . It follows that and cannot be adjacent in , for if they were then the edge of between those two vertices and the edge between and would intersect. Hence . For similar reasons we see that when and . ∎
Lemma 12.4 says that if there is an entry 1 at position , then there are no other entries 1 either upwards and leftwards, or downwards and rightwards, of that position. We also observe that there are only finitely many entries 1 in any row or column of , for if there were infinitely many then the vertices of one of the paths and would have to accumulate at a single vertex of the other.
A consequence of these observations is that a positive -tiling has infinitely many entries 1 if and only if there are increasing sequences of positive integers and such that either for , or for . The class of -tilings satisfying both these conditions was studied in [HoJo2013].
Theorem 12.5**.**
Let be a positive -tiling. The tiling has infinitely many entries if and only if either and are equal and irrational, or and are equal and irrational.
The proof is similar to that of Theorem 11.3, so we hasten through some of the details.
Proof.
Suppose first that has infinitely many entries 1. By Lemma 12.4 (and the fact that there can be only finitely many 1s in any row or column) we see that there are increasing sequences of positive integers and such that either for , or for . In the former case the sequences and must converge to the same irrational limit, so and are equal and irrational. In the latter case and are equal and irrational.
For the converse, suppose that and are equal to some irrational (the other case can be dealt with in a similar manner). Since is irrational, we can find infinitely many pairs of neighbouring vertices of that lie on either side of and accumulate at . The paths and must pass through almost every one of these vertices, so we can find increasing sequences of positive integers and with , for . Then , for each positive integer , so has infinitely many entries . ∎
References