Rigidity of quasicrystallic and -circle patterns
The uniqueness of the orthogonal -circle patterns as studied by Bobenko and Agafonov is shown, given the combinatorics and some boundary conditions. Furthermore we study (infinite) rhombic embeddings in the plane which are quasicrystallic, that is they have only finitely many different edge directions. Bicoloring the vertices of the rhombi and adding circles with centers at vertices of one of the colors and radius equal to the edge length leads to isoradial quasicrystallic circle patterns. We prove for a large class of such circle patterns which cover the whole plane that they are uniquely determined up to affine transformations by the combinatorics and the intersection angles. Combining these two results, we obtain the rigidity of large classes of quasicrystallic -circle patterns.
Circles, especially circle packings and circle patterns, have successfully been used over the past years to define and study discrete analogs of classical smooth objects. In particular, this approach leads to discrete holomorphic mappings, for example discrete analogs of the power functions , and to discrete holomorphic function theory. See for example [23, 7, 8, 11] for some of the contributions to the theory of circle patterns and  for results on circle packings.
In this article we focus on circle patterns which are characterized by a given combinatorics specifying which circles should intersect and by the corresponding intersection angles. Thus we associate to a circle pattern a pattern of kites corresponding to intersecting circles, see Figures 1 (right) and 6. A particularly suitable source for the required knowlegde on circle patterns and their relations to consistency, integrability, and discrete holomorphic functions is the textbook  in discrete differential geometry. Our main results can roughly be summarized as follows: The combinatorics (together with suitable intersection angles and boundary conditions, if necessary) determines the geometry of the circle pattern. This rigidity result can be interpreted as a discrete version of Liouville’s Theorem in complex analysis.
Our first result concerns the uniqueness of circle patterns which are discrete analogs of the power functions for as defined in [7, 4, 2]. Here, the square grid combinatorics, the orthogonal intersection angles, and the two boundary lines of a sector uniquely determine the geometry of the pattern, see Figure 2 for an illustration. The rigidity of orthogonal -circle patterns was only known for rational , see .
Furthermore, we consider the case of a circle pattern which covers the whole complex plane and for which the radii of all circles are equal and the interiors of different kites are disjoint. Then the corresponding kites form a rhombic embedding. Also, assume that there are only finitely many different edge directions of the kites. Such rhombic embeddings are called quasicrystallic . Additionally, orient an edge and consider a line perpendicular to the edge. Move this line parallelly in positive and in negative direction along . We suppose that in both cases this moving line intersects infinitely often edges parallel to . We assume that this property is true at least for two edges with linearly independent directions. Such rhombic embeddings of the plane are for example given by Penrose tilings, see for example Figure 6. We show that any other embedded circle pattern with the same combinatorics and intersection angles of the kites is the image of this embedding by an affine transformation.
Rigidity of some classes of infinite circle patterns of the plane have already been studied. Schramm  considers square grid combinatorics and orthogonal intersection angles. As an essential step of our proof we generalize his result to square grid circle patterns with regular intersection angles and . He  studies disk triangulation graphs and exterior intersection angles in which does not cover the class of isoradial quasicrystallic circle patterns defined above.
As observed in  isoradial quasicrystallic rhombic embeddings can be used to define corresponding quasicrystallic -circle patterns, see also . Examples of quasicrystallic -circle patterns are shown in Figures 7, 9, and 11. They have been created using software developed by Veronika Schreiber for her diploma thesis . Our rigidity result for orthogonal -circle patterns is also generalized for large classes of quasicrystallic -circle patterns.
There is some more literature concerning rigidity for infinite planar circle packings, that is configurations with non-overlapping touching circles, see [21, 15, 22, 20]. Our rigidity proofs adapt some of the ideas which have been used for packings. In particular, we apply discrete potential theory.
This paper is organized as follows. First we introduce terminology and present useful facts about circle patterns. We especially focus on regular circle patterns with square grid combinatorics. Then we recall in Section 4 the definition and some properties of the orthogonal -circle patterns for as studied in [4, 1, 2] and prove their rigidity. In Section 5 we introduce quasicrystallic circle patterns and prove uniqueness for a class of these patterns. Finally, we recall the definition and some facts about quasicrystallic -circle patterns for as studied in [3, 9, 11] and prove rigidity for certain classes of these patterns. A more detailed version of the results can be found in .
2 Circle patterns
In this section we focus on a definition and some useful properties of circle patterns. We describe circle patterns using combinatorial data and intersection angles.
The combinatorics are specified by a b-quad-graph , that is a strongly regular cell decomposition of a domain in possibly with boundary such that all 2-cells (faces) are embedded and counterclockwise oriented. Furthermore all faces of are quadrilaterals, that is there are exactly four edges incident to each face, and the 1-skeleton of is a bipartite graph. We always assume that the vertices of are colored white and black. To these two sets of vertices we associate two planar graphs and as follows. The vertices are all white vertices of . The edges correspond to faces of , that is two vertices of are connected by an edge if and only if they are incident to the same face. The dual graph is constructed analogously by taking as vertices all black vertices of . is called simply connected if it is the cell decomposition of a simply connected domain of and if every closed chain of faces is null homotopic in .
For the intersection angles, we use a labelling of the faces of . By abuse of notation, can also be understood as a function defined on or on . The labelling is called admissible if it satisfies the following condition at all interior black vertices :
Let be a b-quad-graph with associated graph and let be an admissible labelling. An (immersed planar) circle pattern for (or ) and are an indexed collection of circles in and an indexed collection of closed kites, which all carry the same orientation, such that the following conditions hold.
If are incident vertices in , the corresponding circles intersect with exterior intersection angle . Furthermore, the kite is bounded by the centers of the circles , the two intersection points, and the corresponding edges, as in Figure 1 (right). The intersection points are associated to black vertices of or to vertices of .
If two faces are incident in , the corresponding kites share a common edge.
Let be the faces incident to an interior vertex . Then the kites have mutually disjoint interiors. The union is homeomorphic to a closed disk and contains the point corresponding to in its interior.
The circle pattern is called embedded if all kites of have mutually disjoint interiors. The circle pattern is called isoradial if all circles of have the same radius.
Note that we associate a circle pattern to an immersion of the kite pattern corresponding to where the edges incident to the same white vertex are of equal length. The kites may also be non-convex and can be constructed from a (suitable) given set of circles and from the combinatorics of .
As all circle patterns considered in this article will be planar and immersed, the notion “circle pattern” will include these properties in the following.
For our study of a circle pattern we will use the radius function which assigns to every vertex the radius of the corresponding circle . The index will be dropped whenever there is no confusion likely. The following proposition specifies a condition for a radius function to originate from a planar circle pattern, see  for a proof.
Let be associated to a b-quad-graph and let be an admissible labelling.
Suppose that is a planar circle pattern for and with radius function . Then for every interior vertex we have
and the branch of the logarithm is chosen such that .
Conversely, suppose that is simply connected and that satisfies (2) for every . Then there is a planar circle pattern for and with radius function . This pattern is unique up to isometries of .
For the special case of orthogonal circle patterns with the combinatorics of the square grid, there are also other characterizations, see for example .
Note that is the angle at of the kite with edge lengths and and angle , as in Figure 1 (right). Equation (2) is the closing condition for the closed chain of kites which correspond to the edges incident to . This corresponds to condition (3) of Definition 2.1.
For further use we mention some properties of , see [26, Lemma 2.2].
The derivative of is .
The function satisfies the functional equation .
If there exists an isoradial circle pattern, we can obtain another circle pattern from a given radius function.
Let be a graph constructed from a b-quad-graph and let be an admissible labelling. Suppose that there exists an isoradial circle pattern for and . Let be the radius function of a planar circle pattern for and . Then there is a circle pattern for and with radius function .
Let be a graph constructed from a b-quad-graph and let be an admissible labelling. Suppose that and are planar circle patterns for and with radius functions and respectively. Define a comparison function by
Here or is defined to be the rotation angle or the rotation respectively of the edge-star at when changing from the circle pattern to . Note that is the scaling factor of the circle corresponding to . Then satisfies the following Hirota Equation for all faces .
Here and are the black and white vertices incident to and and are the directed edges. Thus equation (4) is the closing condition for the kite corresponding to the face . Furthermore, the Hirota Equation is 3D-consistent; see Sections 10 and 11 of  or  for more details. This property will be used in Section 5.
3 SG-circle patterns
In this paper we are particularly interested in the special case of regular circle patterns with square grid combinatorics. First, we fix some notation. Let be the regular square grid cell decomposition of the complex plane, that is the vertices are and the edges are given by pairs of vertices with and . The 2-cells are squares for . As is a b-quad-graph, the vertices
are assumed to be colored white. As above, and its dual are defined as the associated graphs to . Furthermore, with and denotes the subgraph of all vertices with combinatorial distance at most from in .
Let be a fixed angle. Define the following regular labelling on . Let be an edge connecting the vertices . Without loss of generality, we assume that . Then
If is a subgraph of , a circle pattern for and is called -circle pattern. The choice leads to orthogonal -circle patterns as considered by Schramm in .
Theorem 3.1 (Rigidity of -circle patterns).
Suppose that is an embedded planar circle pattern for and . Then is the image of a regular isoradial circle pattern for and under a similarity.
The proof is a suitable adaption of the corresponding proof for orthogonal -circle patterns given by Schramm using suitable Möbius invariants, see [23, Theorem 7.1] or . This adaption needs the following generalization of the Ring Lemma of , which is also useful in the following.
Let be the radius function of an embedded circle pattern for and . There is a constant , independent of , such that for there holds
Assume the contrary. Then there is a sequence of embedded circle patterns for and such that and as for some . Without loss of generality we assume that . We also may assume that the circle corresponding to the vertex and the intersection point corresponding to are fixed for the whole sequence. Then there is a subsequence such that all the circles converge to circles or lines, that is converge in the Riemann sphere . Now equation (2) implies that there exist some kites which intersect in the limit in their interiors. But this is a contradition to the embeddedness of the sequence. ∎
If the number of surrounding generations is big enough, there is the following useful estimation on the quotient of radii of incident vertices.
There is an absolute constant such that the following holds.
Let be a subgraph of and let be an embedded circle pattern for and with radius function . Let be a vertex and suppose that , that is contains generations of around , for some . Then for all vertices incident to there holds
4 Uniqueness of orthogonal -circle patterns
An orthogonal circle pattern with the combinatorics of the square grid associated to the map was introduced by Bobenko in . Further development of the theory is due to Agafonov and Bobenko [4, 1, 2].
4.1 Definition and useful properties
Let . A map is called discrete conformal if all its elementary quadrilaterals are conformal squares, i.e. their cross-ratios are equal to :
Here and below we abbreviate .
A discrete conformal map is called embedded if the interiors of different elementary quadrilaterals are disjoint.
Note that the definition of a discrete conformal map is Möbius invariant and
is motivated by
the following characterization for smooth mappings:
A smooth map is called conformal (holomorphic or antiholomorphic) if and only if for all there holds
In order to construct an embedded discrete analog of the following approach is used. Equation (7) can be supplemented with the nonautonomous constraint
The asymptotics of the constraint (8) for and the properties and of the holomorphic mapping motivate the following definition of the discrete analog.
From this definition, the properties and are obvious for all . Furthermore, the discrete conformal map from Definition 4.2 determines an -circle pattern. Indeed, by Proposition 1 of  all edges at the vertex with have the same length and all angles between neighboring edges at the vertex with are equal to . Thus, all elementary quadrilaterals build orthogonal kites and for any with the points lie on a circle with center . Therefore, we consider the sublattice and denote by the quadrant
where . Two vertices are connected by an edge if and only if .
If denotes the radius function corresponding to the discrete conformal map for some , then it holds that
for all .
For , the discrete conformal maps given by Definition 4.2 are embedded. Consequently, the corresponding circle patterns are also embedded.
In the following section and in Section 6 we continue to use the notation of this section. In particular the radius function is denoted by and we have the normalization .
4.2 Uniqueness of the orthogonal -circle patterns
This section is devoted to the proof of following uniqueness result.
Theorem 4.4 (Rigidity of orthogonal -circle patterns).
For the infinite orthogonal embedded circle pattern corresponding to is the unique embedded orthogonal circle pattern (up to global scaling) with the following two properties.
The union of the corresponding kites of the -circle pattern covers the infinite sector with angle .
The centers of the boundary circles lie on the boundary half lines and .
Our proof uses results of discrete potential theory or of the theory of random walks which can be found in standard textbooks, for example by Doyle and Snell  or by Woess . We recall some basic terminology and notation and cite adapted versions of a few theorems which will be useful for our argumentation.
By abuse of notation, we denote by the points with as well as the graph with vertices at these points and edges if . The meaning will be clear from the context.
Consider the network with conductances and resistances on the undirected edges . Then a transition probability function is given by
The stochastic process on given by this probability function is a reversible random walk or a reversible Markov chain on . The simple random walk on is given by specifying for all edges which leads to if .
Denote by the probability that a random walk starting at any point will never return to this point. The network () is called recurrent if (and transient otherwise). Note that , where denotes the effective resistance from a point to infinity.
The simple random walk on is recurrent.
Let and be two networks with conductances and on the edges. If for all edges , then the recurrence of implies the recurrence of .
A function is called superharmonic (subharmonic) with respect to the probability function or with respect to the conductances if for every vertex we have ().
The following proposition shows that the quotient of the radius functions of two orthogonal -circle patterns is subharmonic with respect to suitably chosen conductances. As the statement is a special case of Proposition 6.5 below, we omit the proof.
Consider two orthogonal circle patterns for . Denote the radii by and respectively, where and denote the radii of the inner circles. Then
Our proof of rigidity is based on the following property of superharmonic functions on recurrent networks.
Theorem 4.7 ([28, Theorem (1.16)]).
A network is recurrent if and only if all nonnegative superharmonic functions are constant.
Proof of Theorem 4.4..
Let and denote by the radius function of the embedded -circle pattern with . Let denote the radius function of an embedded orthogonal circle pattern with the same combinatorics and the same boundary conditions (orthogonal boundary circles to the half lines and ). Without loss of generality we assume same normalization . This can always be achieved by a suitable scaling. In the following, we will show that the radius functions and take the same values on all of . This implies that both circle patterns coincide.
As both circle patterns are embedded, Lemma 3.3 implies that for some constant and
holds for all radii for vertices of the th generation away from the origin and their incident vertices . Here, vertices belong to the th generation if their combinatorial distance in to the origin is . For estimation (11) we have also used that the reflection of the circle pattern in one of the boundary lines or also leads to an embedded orthogonal -circle pattern. The same reasoning applies to the radii of the -circle pattern, so
for with the same constant . Estimations (11) and (12), the boundary conditions and a suitable adaptation of the Ring Lemma 5.11 for circles of generation two and three from the origin imply that there is a constant such that
for all incident vertices and .
We now consider two undirected networks and as follows. On the edges of we define two conductance functions and by
where the edge connects the vertices . Estimations (13) imply that both positive functions and are uniformly bounded away from (and from infinity). These two conductance networks on can be continued to all of by reflection in the lines . From Theorem 4.5 we deduce that both networks and are reccurent.
Consider the following positive functions on
By Proposition 4.6 these functions are subharmonic. Using the boundary conditions of the circle patterns, this remains true if and are continued to all of using reflection. Consequently, and are superharmonic for all constants . If or is bounded from above, we get a positive superharmonic function using the upper bound. Then Theorem 4.7 implies that both functions are constant. Thus and consequently both circle patterns coincide.
Denote by and the maximum of and , respectively, for the set of vertices of the th generation about the origin. As and are subharmonic, they assume their maxima on the boundary. Therefore the functions and are monotonically increasing. The estimations (11) and (12) imply that the quotients of any two radii of one circle pattern in the th generation are bounded from above for , as two vertices in the th generation can be connected by at most edges using only vertices of the th and st generation. So their quotient is bounded by for both radius functions and . Note that with the normalization , the maxima and are bounded from below by . Thus their product
is bounded from above. Here and denote the vertices of the th generation where and assume their maxima, respectively. Therefore and are also bounded. This finishes the proof of uniqueness. ∎
5 Uniqueness of isoradial quasicrystallic circle patterns
The uniqueness result of Theorem 3.1 can be generalized for some classes of quasicrystallic circle patterns. On this basis we will then generalize Theorem 4.4 for some classes of quasicrystallic -circle pattern.
5.1 Quasicrystallic circle patterns and connection to
A rhombic embedding in of a b-quad-graph is an embedding with the property that each face of is mapped to a rhombus. Given a rhombic embedding of , consider for each directed edge the vector of its embedding as a complex number with length one. Half of the number of different values of these directions is called the dimension of the rhombic embedding. If is finite, the rhombic embedding is called quasicrystallic.
Adding circles with centers in the white vertices of the rhombic embedding and radius equal to the edge length reveals the close connection to embedded isoradial circle patterns.
A circle pattern for a b-quad-graph is called a quasicrystallic circle pattern if there exists a quasicrystallic rhombic embedding of and if the intersection angles are taken from this rhombic embedding. The comparison function of the isoradial circle pattern for and the quasicrystallic circle pattern is also called comparison function for .
In the following we will often identify the b-quad-graph with a rhombic embedding of .
The notion “quasicrystallic” is not uniquely defined in literature. Here we adopt the definition given in . Naturally, this property only makes sense for infinite graphs or sequences of graphs with growing number of vertices and edges.
Any rhombic embedding of a b-quad-graph can be seen as a sort of projection of a certain two-dimensional subcomplex (quad-surface) of the multi-dimensional lattice (or of a multi-dimensional lattice which is isomorphic to ). An illustrating example is given in Figure 3.
The quad-surface in can be constructed in the following way. Denote the set of the different edge directions of by . We suppose that and that any two non-opposite elements of are linearly independent over . Let denote the standard orthonormal basis of . Fix a white vertex and the origin of . Add the edges of at the origin which correspond to the edges of incident to in , together with their endpoints. Successively continue the construction at the new endpoints. Also, add two-dimensional facets (quadrilateral faces) of corresponding to the quadrilateral faces of , spanned by incident edges.
A quad-surface in corresponding to a quasicrystallic rhombic embedding can be characterized using the following monotonicity property. For a proof see [9, Section 6].
Lemma 5.3 (Monotonicity criterium).
Any two points of can be connected by a path in with all directed edges lying in one -dimensional octant, that is all directed edges of this path are elements of one of the subsets of containing linearly independent vectors.
Example 5.4 (Quasicrystallic rhombic embedding obtained from a plane).
Let be a two-dimensional plane in . Let denote the standard orthonormal basis of and let . We assume that does not contain any of the segments for . If contains two different segments and , the following construction only leads to the standard square grid pattern . If contains exactly one segment , the construction may be adapted for the remaining dimensions (excluding ). We further assume that the orthogonal projections onto of the two-dimensional facets for are non-degenerate parallelograms. Then we can choose positive numbers such that the orthogonal projections have length 1.
Consider around each vertex of the lattice the hypercuboid , that is the Voronoi cell . These translations of cover . If intersects the interior of the Voronoi cell of a lattice point (i.e. for ), then this point belongs to . Undirected edges correspond to intersections of with the interior of a -dimensional facet bounding two Voronoi cells. Thus we get a connected graph in . An intersection of with the interior of a translated -dimensional facet of corresponds to a rectangular two-dimensional face of the lattice. By construction, the orthogonal projection of this graph onto results in a planar connected graph whose faces are all of even degree ( number of incident edges or of incident vertices). A face of degree bigger than 4 corresponds to an intersection of with the translation of a -dimensional facet of for some . Consider the vertices and edges of such a face and the corresponding points and edges in the lattice . These points lie on a combinatorial -dimensional hypercuboid contained in . By construction, it is easy to see that there are two points of the -dimensional hypercuboid which are each incident to of the given vertices. Choose a point with least distance from and add it to the surface. Adding edges to neighboring vertices splits the face of degree into faces of degree 4.
5.2 Quasicrystallic circle patterns and integrability
Let be a quasicrystallic rhombic embedding of a b-quad-graph. The quad-surface in is important by its connection with integrability. See also  for a more detailed presentation and a deepened study of integrability and consistency.
In particular, a function defined on the vertices of which satisfies some 3D-consistent equation on all faces of can be uniquely extended to the brick
where and . Note that is the hull of . A proof may be found in [9, Section 6]. Let be a quasicrystallic rhombic embedding and let be a quasicrystallic circle pattern with the same combinatorics and the same intersection angles. Denote the comparison function for by as in (3). Since the Hirota equation (4) is 3D-consistent (see Sections 10 and 11 of ) considered as a function on can uniquely be extended to the brick such that equation (4) holds on all two-dimensional facets. Additionally, and its extension are real valued on white points of and have values in for black points of . This can easily be deduced from the Hirota Equation (4).
The extension of can be used to define a radius function for any rhombic embedding with the same boundary faces as .
Let and be two simply connected finite rhombic embeddings of b-quad-graphs with the same edge directions. Assume that and agree on all boundary faces. Let be an (embedded) planar circle pattern for and the labelling given by the rhombic embedding. Then there is an (embedded) planar circle pattern for and the corresponding labelling which agrees with for all boundary circles.
Consider the monotone quad-surfaces and . Without loss of generality, we can assume that and have the same boundary faces in . Thus both define the same brick . Given the circle pattern , define the comparison function for on by (3). Extend to the brick such that condition (4) holds for all two-dimensional facets. Consider on and build the corresponding pattern , such that the points on the boundary agree with those of the given circle pattern . Equation (4) guarantees that all faces of are mapped to closed kites. Due to the combinatorics, the chain of kites is closed around each vertex. Since the boundary kites of are given by which is an immersed circle pattern, at every interior white point the angles of the kites sum up to . Thus is an immersed circle pattern.
Furthermore, is embedded if is, because is an immersed circle pattern, and and agree for all boundary kites. ∎
5.3 Local changes of rhombic embeddings
Let be a rhombic embedding of a finite simply connected b-quad-graph and let be the corresponding quad-surface in . Let be an interior vertex with exactly three incident two-dimensional facets of . Consider the three-dimensional cube with these boundary facets. Replace the three given facets with the three other two-dimensional facets of this cube. This procedure is called a flip; see Figure 4 for an illustration.
A vertex can be reached with flips from if is contained in a quad-surface obtained from by a suitable sequence of flips. The set of all vertices which can be reached with flips, including , will be denoted by .
The quad-surface can also be generalized using a finite sequence of flips. Such an infinite rhombic embedding will be called a plane based quasicrystallic rhombic embedding.
As a generalization of simple flips we define flips for simply oder doubly infinite strips of the following form. See Figure 5 for an illustration.
Let be a simply connected monotone quad-surface. Let be a white vertex. Let be three different edges incident to such that there are two-dimensional faces of incident to and , and to and , respectively. Let and be the intersection angles associated to these faces. Let be the intersection angles associated to the two-dimensional facet of incident to and . Then or . In the first case, consider the half-axis , where is the vector corresponding to the edge and pointing away from as in Figure 5. In the second case, consider the other half-axis . In both cases we may also consider the whole axis