Tiling the Euclidean and Hyperbolic planes with ribbons
TTiling the Euclidean and Hyperbolicplanes with ribbons
Benedict Kolbe and Vanessa Robins ∗ , Institute of Mathematics, Technische Universit¨at Berline-mail: [email protected]
Research School of Physics, The Australian National University, Canberrae-mail: [email protected]
Abstract:
We describe a method to classify crystallographic tilings of theEuclidean and hyperbolic planes by tiles whose stabiliser group containstranslation isometries or whose topology is not that of a closed disk. Wetackle this problem from two different viewpoints, one with constructivetechniques to enumerate such tilings and the other from a viewpoint ofclassification. The methods are purely topological and generalise Delaney-Dress combinatorial tiling theory. The classification is up to equivariantequivalence and is achieved by viewing tilings as decorations of orbifolds.
Keywords and phrases:
Delaney-Dress tiling theory, Ribbon tiles, Friezegroups.
1. Introduction
Patterns built from repeating motifs appear in all cultures and have long beenstudied in art, mathematics, engineering and science. Most mathematical workhas focussed on patterns in the Euclidean plane (the book “Tilings and Pat-terns” by Gr¨unbaum and Shephard [11] contains a comprehensive survey of thefield up to the mid 1980s) but the importance of hyperbolic geometry as a modelfor natural forms is increasingly recognised [16, 24, 18, 29]. An example thatinspires the work in this paper is the discovery that star co-polymer systemsconsisting of three mutually immiscible arms can self-assemble into structuresmodelled by stripes on the gyroid triply periodic minimal surface [19, 1]. Thegyroid surface has genus three in its smallest side-preserving translational unitcell, and therefore has the hyperbolic plane as its simply-connected coveringspace. Its 3d space-group symmetries induce a non-euclidean crystallographicgroup generated by hyperbolic isometries that are known explicitly [28, 27].Stripe patterns on the gyroid lift via the covering map to tilings of the hy-perbolic plane by infinitely long strips, or ribbons. The defining property of aribbon tile is the existence of a translation isometry that maps a given tile backonto itself. This paper considers the general question of how to describe andenumerate crystallographic tilings of the Euclidean and hyperbolic planes byribbon tiles. ∗ Corresponding author supported by ARC Future Fellowship FT140100604.1 imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 a r X i v : . [ m a t h . G T ] A p r olbe and Robins/Ribbon tilings The Euclidean case of striped patterns is described in Section 6.5 of [11] citingearlier work by Wollny [33]; there are 26 distinct types of crystallographic ribbontilings of the Euclidean plane. To obtain corresponding results for the hyperbolicplane we require different mathematical techniques to those used by Gr¨unbaumand Shephard. Our approach derives from the perspective of Dress et al. whodeveloped the field of combinatorial tiling theory [5, 6] and from the classificationof 2d discrete groups of isometries via their quotient spaces [23, 32, 31, 21, 2].Combinatorial tiling theory treats periodic (crystallographic) tilings of a simplyconnected space where the tiles are topological disks and defines an invariantcalled the Delaney-Dress symbol or
D-symbol as a coloured weighted graph.From the D-symbol it is possible to reconstruct the tile shapes and adjacenciesand the isomorphism class of the symmetry group for the tiling.In the 2d setting, a D-symbol encodes a finite triangulation derived from thetiling. The underlying space of the triangulation is the quotient of the planeby a discrete group of isometries that preserve the tiling. This quotient spaceis a 2-orbifold and can be viewed as a compact surface with (possibly) a finitenumber of boundary components and a finite number of isolated marked points.Although the full theory of D-symbols does not directly generalise to our set-ting of tilings by ribbons, the correspondence between crystallographic patternsand decorations of 2-orbifolds certainly does. The main issues to overcome arecharacterising the possible stabiliser subgroups for unbounded ribbon tiles, andconstructing a triangulation from a given pattern. For Euclidean tilings by rib-bons the possible stabiliser groups are the seven frieze groups; for the hyperboliccase, infinitely many non-euclidean frieze groups are also possible.Previous work related to this paper includes Huson’s paper on tile-transitivepartial tilings of the Euclidean plane [15], and the exploration of crystallographicline and tree patterns in the hyperbolic plane by Hyde et al. [17, 8, 9, 7]. Inthis paper, we assume the tilings are locally finite and acted on by a discretegroup of isometries of the plane with compact fundamental domain. Definitionsand notation for these groups of isometries are given in Section 2.1, and forcombinatorial tiling theory in Section 2.2. Results characterising the existenceand structure of ribbon tiles are in Section 3. Algorithms for enumerating andclassifying tilings are described in Section 4, with examples in Section 5.
2. Definitions and Preliminaries
Let X be either the Euclidean ( E ) or hyperbolic ( H ) plane, and let Γ be adiscrete group of isometries of X having a compact fundamental domain. If X = E then Γ is one of the 17 wallpaper groups of crystallography. If X = H ,then Γ is a NEC group (non-Euclidean crystallographic group). We identifythe isomorphism class of a group using Conway’s orbifold symbol [3], a highly-readable version of Macbeath’s group signature [23], as described below.For the purposes of this paper a 2-orbifold, O = X / Γ, is a quotient spaceobtained by identifying points of X under the action of Γ. That is, x ∼ y if imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings y = γx for some γ ∈ Γ. The difference between X / Γ as a topological space andas an orbifold is that the full orbifold structure retains the metric informationcarried by the particular isometries of Γ and an atlas of charts compatible withthe Γ action. We require both the topological view point and orbifold structureof the quotient space in this paper, and will use the script notation O for theorbifold with the additional structure and plain O for its underlying topologicalspace.It is well-known [31] that 2-orbifolds have the topology of a finite-area 2-manifold with a finite number of boundary components. Boundaries in a 2-orbifold arise from the fixed lines of reflection isometries. Other special pointsarise as the centres of rotational isometries; these are called cone points if theylie in the interior of the orbifold, and corner points if they lie on a boundary. Thebranching number, N , of a cone or corner point is the order of the rotationalisometry, σ , that fixes that point i.e. σ N = id . The boundaries, corner andcone points are collectively referred to as the singular locus of the orbifold. Thetopology of a 2-orbifold ( O ) is therefore specified by a symbol as follows:1. The number of handles, h , if the orbifold is orientable, or the number ofcross-caps, k , if non-orientable. Handles are denoted by ◦ at the beginningof the orbifold symbol. Cross-caps are denoted by × at the end of theorbifold symbol.2. The branching number for each cone point, listed in arbitrary order afterany handles.3. The number of boundary components, q . Each boundary component isrepresented by a ∗ in the symbol. Branching numbers for the corner pointslying on each boundary component are listed in cyclic order, such that eachboundary component has a consistent orientation for the manifold. Theordering of the boundary components is arbitrary.As simple examples, the group of isometries for a tiling of E by squares meet-ing four to a corner is ∗ ( p m in Hermann-Mauguin notation), but for aEuclidean pattern with only translational symmetries it is ◦ ( p E , H , or S , except for the symbols A , ∗ A , AB , and ∗ AB , with A (cid:54) = B . Moreover, the plane geometry associated with an orbifold can be deducedby computing a curvature-related quantity (the orbifold Euler characteristic)directly from the group symbol.Other discrete groups of isometries will be important when we discuss theinternal symmetries of a tile as defined in the following section. For a boundedtile T , homeomorphic to a disk, the possible symmetry groups include thatgenerated by a single reflection ( D ) and those that fix a single point. Thelatter are well known as the cyclic and dihedral groups of order N ≥ C N = (cid:10) q | q N = id (cid:11) D N = (cid:10) r , r | r = id, r = id, ( r r ) N = id (cid:11) When these groups are viewed as acting on the tile T , we can form the quo- imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings tient space T /C N or T /D N and describe the topology of these quotient spacessimilarly to the orbifold symbol above. This yields what Conway et al. [2] call signatures for rosette patterns: N • and ∗ N • for the cyclic and dihedral groupsrespectively. The symbol • represents for us a section of tile boundary in thequotient space. Note that the group generated by a single reflection isometry, D = (cid:10) r | r = id (cid:11) is abstractly isomorphic to C , but its quotient space has thesignature ∗• .As we want to classify tilings by ribbons, we also need to consider, for ex-ample, isometries of a tile T that is homeomorphic to [0 , × R . The possiblediscrete symmetry groups for such a tile are D , D , C or one of the sevenfrieze groups. Again, we can specify the quotient space structure of T /G using adescriptive signature or symbol. In Table 1 we give the signatures (from [2]) andHermann-Mauguin (IUCr) name of each of the frieze groups. In these signaturesthe ∗ symbol still represents a single boundary component of T /G and in oursetting the ∞ symbol represents a segment of tile boundary in T /G . So the sig-nature ∗∞∞ implies that
T /G is a disk with a mirror boundary interrupted bytwo tile boundary segments and so combinatorially a quadrilateral. Note thatin other contexts, the ∞ symbol can represent an orbifold puncture as might begenerated by a parabolic isometry of H . The different contexts are just otherways of obtaining a geometric realisation of the same abstract group. Table 1
Signature of
T/G and IUCr name for the seven frieze groups, the index of the translationisometry in G , and a presentation that makes the translation isometry t explicit in eachcase. T/G name index group presentation ∞∞ p1 (cid:104) t (cid:105) ∞ x p11g (cid:10) g, t | g = t (cid:11) ∞∗ p11m (cid:10) r, t | r , rtr = t (cid:11) ∗∞∞ p1m1 (cid:10) r , r , t | r , r , r r = t (cid:11) ∞ p2 (cid:10) q , q , t | q , q , q q = t (cid:11) ∗ ∞ p2mg (cid:10) r, q, t | r , q , ( qr ) = t (cid:11) ∗ ∞ p2mm (cid:10) r , r , r , t | r , r , r , ( r r ) , ( r r ) , r r = t (cid:11) It is important to note that the above rosette and frieze groups can be realisedusing isometries of either the Euclidean or hyperbolic plane. If a rosette or friezegroup, G , occurs as the subgroup of a wallpaper or NEC group, Γ, then it hasinfinite index, i.e. there are infinitely many cosets γG . The general question ofcharacterising the subgroups of infinite index in NEC groups is covered in [12, 13]where it is shown, as a special case, that if the subgroup has no reflectionelements then it must be a free product of cyclic groups (of finite or infiniteorder). So, for example, ∞ is simply the free product of two copies of C . Theother frieze groups are given in Table 2.In the hyperbolic plane, we will also naturally encounter unbounded simplyconnected tiles with branching structure homeomorphic to a neighbourhood ofan infinite tree embedded in H . We call such tiles branched ribbons , examplesare given in Section 5. The isometries of such a tile are isomorphic to a groupaction on a tree, and are covered by the theory of Bass-Serre [30]. The simplest imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings Table 2
Frieze groups and their presentation as per results in [13]. The symbols t and g aregenerators of infinite order, with t having a geometric interpretation as a translation and g as a glide. The other generators use r and q which can be interpreted geometrically asreflections and rotations respectively. G group presentation ∞∞ C ∞ = (cid:104) t (cid:105) ∞ x C ∞ = (cid:104) g (cid:105) ∞∗ C ∗ C ∞ = (cid:10) r, g | r (cid:11) ∗∞∞ C ∗ C = (cid:10) r , r | r , r (cid:11) ∞ C ∗ C = (cid:10) q , q | q , q (cid:11) ∗ ∞ C ∗ C = (cid:10) r, q | r , q (cid:11) ∗ ∞ C ∗ C ∗ C = (cid:10) r, q , q | r , q , q (cid:11) examples of group actions on trees are those that have a line segment as fun-damental domain. These groups are a free product with amalgamation of thesubgroups that fix the vertices, amalgamated via the subgroup that fixes theedge (the oriented line segment). For example, if the line segment generatingthe tree has end points on rotation centers of order A and B , and the edge groupis trivial, then the group is G = C A ∗ C B and T /G has the signature AB ∞ . Combinatorial tiling theory describes the combinatorial structure of tilings of asimply connected space for which each tile is homeomorphic to a closed andbounded disk. The definitions below follow those given for two-dimensionalspaces in [14]. A set T of topological disks in X is called a tiling if every point x ∈ X belongs to at least one tile T ∈ T and every two tiles T and T of T havedisjoint interior. All tilings in this paper will be assumed to be locally finite,i.e. any compact disk in X meets only a finite number of tiles. The verticesand edges of a tile are defined topologically rather than using the geometry ofstraight lines and corners. So, a vertex is a point that is contained in at leastthree tiles, and an edge is a connected segment of tile boundary joining twovertices.Let T be a tiling of X and let Γ be a discrete group of isometries. If T = γ T := { γT | T ∈ T } for all γ ∈ Γ then we call the pair ( T , Γ) an equivarianttiling . Two tiles T , T ∈ T are equivalent or symmetry-related if there exists γ ∈ Γ such that γT = T . The orbit of a tile is the subset of T given byimages of T : Γ .T = { γT for γ ∈ Γ } . Given a particular tile T ∈ T , the stabilizersubgroup Γ T is the subgroup of Γ that fixes T , i.e. Γ T = { γ ∈ Γ | γT = T } . A tileis called fundamental if Γ T is trivial and we call the whole tiling fundamental ifthis is true for all tiles. An equivariant tiling is called tile - k - transitive , when k is the number of equivalence classes (i.e. distinct orbits) of tile under the actionof Γ. We can also study the action of Γ on tile edges and vertices and defineedge- and vertex- k -transitivity similarly. Note that the above definitions do notrequire Γ to be the maximal symmetry group for the tiling T .Two tilings ( T , Γ ) and ( T , Γ ) of a simply connected space X are equiv- imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings ariantly equivalent if there is a homeomorphism, φ , of X such that φ ( T ) ∈ T for all T ∈ T and such that φ induces a group isomorphism of Γ onto Γ byΓ = φ Γ φ − . A natural question is whether there is an invariant that detectswhen two tilings are equivariantly equivalent. Dress et al. , [5] show that a com-plete invariant is indeed possible for tilings of simply connected manifolds. Theinvariant, called the D-symbol, consists of a graph that records adjacencies be-tween tiles and their faces, augmented by weights that encode the group actionof Γ on T . The D-symbol can be interpreted as encoding a simplicial structureon an orbifold, obtained from barycentric subdivision of the tiling. For the 2dcase this means a triangulation of a 2-orbifold where each 2-simplex spans a tilecentre-point, edge mid-point and tiling vertex. This structure is exploited in [14]to achieve a fully algorithmic approach to the enumeration and identificationup to equivariant equivalence of 2d tilings of S , E and H .A fundamental tile-1-transitive equivariant tiling ( T , Γ) has a single type oftile, T , that is a fundamental domain for Γ. Conversely, any fundamental do-main for Γ homeomorphic to a disk also gives rise to such a tiling. As establishedby Wilkie in [32] such a tiling corresponds to a presentation for Γ, with each edgeof T corresponding to a generator and each vertex to a relation. If e i = T ∩ T i ,then the edge-generator is an element, [ e i ] ∈ Γ, such that T i = [ e i ] T . The re-lation at vertex v is found by listing the generators associated with each edgecrossing when making a clockwise circuit around v . See [32] for further details.The papers by Lucic et al. [20, 21, 22] present a method for enumeratingthe different combinatorial forms of disk-like fundamental domains for crystal-lographic groups. In particular, the authors establish that vertices and edgesof a tiling ( T , Γ) map via the group action onto vertices and edges of a graph C = ( V, E ) embedded in the quotient space O = X / Γ. In general, vertices of C come from vertices of T , except in the case that an edge midpoint in T is acone point of order 2. Such an edge of T maps onto an edge of C with a vertexof degree 1 on the cone point. The graph C embedded in O has the followingproperties:1. O \ C is an open disk.2. Each cone point is a vertex of C with at least one incident edge in C .3. Each (mirror) boundary component of O lies in a subgraph of C .4. Each corner point is a vertex of C with at least two incident edges.5. Let ˜ O be the closed surface of genus g obtained from O by capping eachboundary component of O with a disk. Then C is contractible in ˜ O to thegraph ˜ C with one vertex and 2 g loops if O is orientable and g loops if O is non-orientable. In particular, each subgraph C i that belongs to the i -thboundary component of O is a contractible loop in ˜ O .6. Any vertex of C that does not lie on a cone point, corner point or boundarymust have at least three incident edges in C .See [21, 22] for representative general figures, and figure 1 for a simple Euclideanexample.Another approach to enumerating fundamental tile-1-transitive tilings us-ing a combinatorial requirement on D-symbols is given by Huson in [14]. The imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings same paper then shows that tile- k -transitive fundamental tilings can be derivedfrom those of transitivity ( k −
1) using the operation of tile splitting and non-fundamental tile- k -transitive tilings are obtained from fundamental ones by tileglueing . We describe these operations below in terms of modifications to C , thegraph of tile edges in O .Given a tile- k -transitive fundamental tiling, let C be its corresponding graphon O . Tile-splitting adds a new segment e to C such that O \ ( C ∪ e ) is the unionof k + 1 disks.Now, suppose we have a tile- k -transitive tiling with j ≤ k classes of fun-damental tile. A fundamental tile is identified as a component of O \ C thatcontains no part of the singular locus of O . Tile glueing erases at most twoedges from C to get C (cid:48) so that C (cid:48) is connected, O \ C (cid:48) is still the union of k disks, and the tiling has ( j −
1) classes of fundamental tile. The edges of C thatcan be erased must be incident only to one transitivity class of fundamental tileand are of three types:1. An edge of C that has a vertex of degree 1 at a cone point of order N .This glues N copies of a fundamental tile into one new one with stabilisergroup N • .2. A pair of edges of C that lie in a mirror boundary and meet at a ver-tex of degree 2 on a corner point of order N . This glues 2 N copies of afundamental tile into one new one with stabiliser group ∗ N • .3. A single edge of C that is a segment of mirror boundary and has verticesof degree at least 3 (or degree 2 on a corner point). This glues two copiesof a tile together into one with stabiliser group ∗• .As discussed in [14], a sequence of tile splits and glues can sometimes leadto a tile whose interior is a disk, but whose closure is not. Such tiles can beidentified algorithmically from the combinatorics of the D-symbol as describedin that paper.In the following sections, we develop the mathematical groundwork requiredto characterise edge erase operations that lead to unbounded ribbon and branched-ribbon tilings.
3. Orbifold paths and tile glueing
We will study the relationship between closed paths in the orbifold O = X / Γ,their lifts in X (i.e. the Euclidean or hyperbolic plane), and equivariant tilingsthat contain unbounded tiles. Firstly: Definition 3.1.
A (possibly branched) ribbon tiling of X is a countable set T of connected closed domains, T i , such that every point x ∈ X belongs to at leastone tile, all tiles have pairwise disjoint interiors, and such that any boundeddisk in X intersects finitely many tiles.An equivariant ribbon tiling is one that is mapped to itself by a discretegroup of isometries of X . A simple example of a ribbon tiling that is not an imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings example of a classical tiling is the trivial tiling, where there is only one tile thatis all of X . Other examples are the partial tilings enumerated by Huson in [15],if we treat his complementary regions as ribbon tiles.Below, we denote the singular locus of O (and O ) by Σ, and the projectionmap as p : X → O . Definition 3.2. An orbifold loop in O is built from a sequence of paths { α i } ki =1 :[ t i − , t i ] → X where 0 = t < t < · · · < t k = 1 such that the α i project to aloop in O , i.e. p ( α i ( t i )) = p ( α i +1 ( t i )) ∀ i ∈ { , ..., k − } and p ( α k ( t k )) = p ( α (0)).Furthermore, we require each segment α i ([ t i − , t i ]) to be contained in a singleorbifold coordinate patch, and attach to each t i , i = 1 , . . . , k , an element γ i ∈ Γthat corresponds to a coordinate change map from the patch containing α i tothat containing α i +1 , with γ k being a coordinate change from α k to α .The group elements γ i that lift the coordinate changes at the points wherethe α i fit together let us distinguish the situation where an orbifold path crossesa mirror line in X ( γ is the reflection) or simply backtracks after touching it ( γ is the identity).We also define a simple orbifold loop as one that has no self-intersections in O and does not pass through any cone or corner points.Two orbifold loops are homotopic if the sequence of paths in X are Γ-equivariantly homotopic. Note that we explicitly allow for concatenation andsplitting of paths, where applicable, i.e. the number of segments, k , in the defi-nition above is not necessarily fixed during a homotopic deformation. Also referto [26] for further details.The orbifold fundamental group π orb ( O , x ), x ∈ O\ Σ, can be defined as the setof orbifold loops based at x ∈ X up to homotopy equivalence in X , where x ∈ p − ( x ). In practice, one can interpret this to mean that a loop in O that windsonce around a cone point of order N cannot be contracted to the point x , buta closed path that winds exactly N times around a neighbourhood of this conepoint (and avoids the rest of Σ) is null-homotopic. Note that the contractibilityof an orbifold loop cannot be seen simply by looking at the corresponding imagecurve in O , in contrast to the classical situation for manifolds.It is known that for our geometric 2-orbifolds, π orb ( X / Γ) (cid:39) Γ in the naturalway. In other words, π orb ( X / Γ) is the group of deck transformations for the branched covering map p : X → O . For proofs and a more detailed account,refer to [26, chapter 13].We now study what happens when we erase a subset of edges from a fundamental-domain tiling. Recall that the edges of a tiling map to an embedded graph C = ( V, E ) in O . Let S = { e , . . . , e k } ⊂ E be the edges to be erased and R = E \ S be the edges that remain. Let C R = ( V R , R ) ⊂ C be the subgraphobtained from C by erasing S and any vertices left isolated. We also want toavoid “dangling ends”, so for all e ∈ R with a vertex v of degree-1 in C R with v / ∈ Σ we add e to S . imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings Theorem 3.3.
Let X = E or H and suppose Γ is a wallpaper or NEC groupSuppose we are given a fundamental tile-1-transitive tiling ( F , Γ) whose edgesmap onto a graph C embedded in O = X / Γ . Let S = { e , . . . , e k } be a subset ofedges of C whose removal avoids dangling ends. Then erasing all preimages ofthese edges from F results in a non-fundamental tile-1-transitive ribbon tiling ( T , Γ) , such that the stabiliser group of each tile T ∈ T is isomorphic to thesubgroup of Γ generated by the erased edges.Proof. The correspondence between edges of a fundamental domain, F ⊂ X ,and generators for a presentation of Γ are established in [32], as summarised inSection 2.2. If f is an edge of F , then f = F ∩ γ ( F ) for an element γ ∈ Γ, andwe use the notation [ f ] for this group element. We also know that each edge f that is not a segment of mirror boundary is glued to another edge f (cid:48) ∈ F (possibly itself) so that [ f (cid:48) ] = [ f ] − . The image of f in O is an edge of C , p ( f ) = p ( f (cid:48) ) = e .Choose a point x ∈ int( F ), and for each e i ∈ C choose a single tile edge f i ∈ p − ( e i ) ∩ cl( F ). Then for each f i , there is a simple orbifold loop α e i with α e i (0) = x and α e i (1) = [ f i ]( x ), such that α e i ([0 , X that intersects the boundary of F in a single point of f i . This follows fromthe fact that F is a topological disk. In the deck-transformation correspondencebetween the orbifold fundamental group and Γ, we then have that [ α e i ] ∼ [ f i ].Now let H be the subgroup of Γ generated by the group elements associatedwith edges e i ∈ S and let T = (cid:83) η ∈ H η ( F ). T is path connected by the followingargument. Each orbifold loop α e i has a connected representative from x ∈ F to[ f i ]( x ) in X , and any other such connected representative of α e i has the end-points γ ( x ) and γ [ f i ]( x ) for some γ ∈ Γ. Therefore, for each η ∈ H , there is apath in X from x to η ( x ) that lies entirely within T , obtained by writing η as aword in [ f ] , . . . , [ f k ], and forming the corresponding concatenation of the liftsof the α e i loops.By their definitions, H is the stabiliser subgroup of the ribbon tile T . If H is finite, then it must be a cyclic or dihedral group, the erased edges must beone of the three cases discussed in Section 2.2, and T will be bounded. If H isinfinite and a proper subgroup, then it is a free product of groups as describedin [12, 13] and T must be unbounded as it is the union of infinitely many distinctcopies of F . The case that H = Γ means T is the whole of X .Next consider the action of an isometry, γ ∈ Γ on a glued tile T . Recall thatthe construction of T began with a particular choice of fundamental domain tile F ⊂ X , and that Γ acts transitively on the tiling F . If γ ∈ H , then γF ∈ T , γη ∈ H for all η ∈ H and so γT = T . If γ / ∈ H , then γF / ∈ T , and in particular, γηF / ∈ T for any η ∈ H , so that γ (int( T )) ∩ int( T ) = ∅ . It follows that for anytwo γ, γ (cid:48) / ∈ H , that either γT = γ (cid:48) T or γ (int( T )) ∩ γ (cid:48) int( T ) = ∅ . So let T bethe union of all distinct images of T . It then follows from the tile-transitivity of( F , Γ), that ( T , Γ) is also a tile-transitive ribbon tiling of X . Lemma 3.4.
The stabiliser subgroup, H , for the non-fundamental tile T , gen-erated by erasing edges as in Theorem 3.3 is infinite if and only if it contains atranslation isometry. imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings Proof.
First note that if H contains a translation isometry then it must beinfinite. And recall that a glide applied twice is a translation.Now assume H is an infinite group of isometries of H . If H is abelian, then itmust be generated by a single isometry of infinite order, i.e., a translation, glideor parabolic rotation (an isometry with a single fixed point at infinity). Thislast case can be ruled out as our tiling group Γ is assumed to have a compactorbifold. If H is non-abelian, then we have a result from [10] that such a groupmust contain a translation.Finally, we consider the case that H is an infinite discrete subgroup of isome-tries of E : H ⊂ O (2) (cid:111) R . If H consists entirely of rotations and/or reflectionisometries, then it must be a discrete subgroup of O (2), and therefore finite.Since we assume H is infinite, not all its elements can be rotations and reflec-tions and we see that it must contain a translation or glide. Lemma 3.5.
Let ( T , Γ) be a tiling obtained via edge deletion from a fundamen-tal tile-1-transitive tiling as in Theorem 3.3. If the stabiliser group H is infinite,then the tile T is simply connected: it is a ribbon or branched ribbon.Proof. From Theorem 3.3, it follows that X is the union of path-connected,unbounded tiles of the form γT for γ ∈ Γ and that all of these tiles have disjointinteriors. If T is not simply connected, then it bounds a region of X that iscovered by isometric copies of T , which is clearly a contradiction.We briefly discuss the tile- k -transitive case. By the results of section 2.2, everytile- k -transitive fundamental tiling comes from splitting an initial fundamentaltile-1-transitive tile into k pieces, each homeomorphic to a disk. Now supposethat 0 < j ≤ k tiles are fundamental, and consider what edges may be erasedso that we obtain a tiling with ( j −
1) fundamental tiles. We want to staywithin the class of tile- k -transitive tilings, so the allowed deletions are restrictedto edges that are incident to only one symmetry class of fundamental tile, asrequired in the classical setting. The construction of the glued tile, T , and itsstabiliser group, H , then proceeds in the same manner as for the tile-1-transitivecase discussed above. Each deleted edge has a simple orbifold loop associatedto it avoiding all other edges, so the resulting tile is path-connected, but notnecessarily homeomorphic to a disk, nor simply connected. If H is finite, thenthe topology of cl( T ) may be that of a closed disk (as before) or a closed diskwith finitely may holes. If H is infinite, then it can have finite or infinite indexin Γ. The first case occurs only if H = Γ. Indeed, the existence of even a singleedge in the graph C embedded in X / Γ results in infinitely many connectededge components in the corresponding tiling in X . We call the tiling in thiscase “patchy.” That is, the tile with stabiliser group H may have bounded holescontaining other tiles; Euclidean examples of these are enumerated in [15]. Inthe case H has infinite index in an NEC group, it must be a free product ofgroups of the form in [13]. The Euclidean case is called “stripey” in [15]: thecomplementary region in those examples is a tile with infinite stabiliser groupand so a ribbon. imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings (a) The Euclidean orb-ifold ∗ which is a spherewith one boundary curve(the red dotted line) andtwo cone points of order2. (b) A slightly deformedstandard fundamental do-main for ∗ . Four fun-damental domains makeup the rectangular trans-lational unit cell shown. (c) The standard funda-mental domain drawn asa graph embedded in theorbifold quotient space. Fig 1: An example of a Euclidean orbifold quotient space, its geometric realisa-tion and edge graph diagram. The vertical lines of (b) lie along the mirrors andthe two inequivalent centres of rotation are marked.
4. Enumeration and classification of branched ribbon tilings
We are now in a position to enumerate crystallographic tilings with ribbon tiles.The steps are as follows:1. Select a symmetry group of interest, and construct its orbifold.2. Enumerate the possible tile-1-transitive fundamental tilings with methodsdescribed in [21, 22] or [14], and represent these as graphs embedded inthe orbifold.3. Systematically delete subsets of edges from the embedded graphs as de-scribed in the section above to derive all tile-1-transitive non-fundamentaltilings, both regular ones with bounded tile and ribbon ones with un-bounded ribbon or branched-ribbon tiles.4. To enumerate tile-2-transitive tilings, apply Huson’s SPLIT algorithm tothe fundamental-domain tile, then systematically delete allowed sets ofedges incident to one tile type, and then the second tile type.5. To build more complex examples, apply successive split operations to fun-damental tiles, followed by edge erasing or tile glueing on each type oftile.Steps 1-3 are illustrated for the wallpaper group pmg with Euclidean orbifold ∗ in figures 1–4.The resulting list of tilings from this enumeration will naturally contain equiv-ariantly equivalent duplicates. For the tile-1-transitive cases where the orbifoldtopology and singular locus is not too complex, it is possible to determine theequivalence classes by eye. For the more complex orbifolds, and tile-k-transitivecases, it is desirable to have a computable invariant. As already known, the D-symbol provides such an invariant for tilings by disks. We now present a method imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings (a) (b) (c)(d) (e) (f) Fig 2: The other combintorial types of fundamental domain for ∗ with (a)triangular, (b) quadrilateral and (c) pentagonal polygon regions. (d)–(f) Thecorresponding edge graphs embedded in the orbifold quotient space. (a) Ribbon tile with sta-biliser ∞∞ . (b) Ribbon tile with sta-biliser ∞∗ . (c) Ribbon tile with sta-biliser ∞ .(d) Ribbon tile with sta-biliser ∗∞∞ . (e) Ribbon tile with sta-biliser ∗ ∞ . Fig 3: By deleting subsets of edges from the four fundamental domain edgegraphs shown in figures 1 and 2, we obtain five possible non-fundamental tilingsof ∗ with frieze group stabilisers. imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings (a) Ribbon tile with sta-biliser ∞∞ . (b) Ribbon tile with sta-biliser ∞∗ . (c) Ribbon tile with sta-biliser ∞ .(d) Ribbon tile with sta-biliser ∗∞∞ . (e) Ribbon tile with sta-biliser ∗ ∞ . Fig 4: These ribbon tiling patterns correspond directly to the quotient graphdiagrams in figure 3. Each drawing shows 2 by 2 translational unit cells in thesymmetry group ∗ . The symmetry of the ribbon in (c) can only be geometri-cally depicted by marking the tiles, e.g. with an ‘L’ motif.for the general case, based on introducing coloured edges to the tiles with infi-nite stabiliser group to obtain a classical tiling-by-disks. As there is more thanone way to introduce the coloured edges, we describe how to find a minimalsymbol, and call this the coloured D-symbol .Given a ribbon tile- k -transitive tiling of the Euclidean or Hyperbolic plane,first identify a crystallographic symmetry group acting on the tiling ( T , Γ) andsome fundamental domain F for the symmetry group. Map this fundamentaldomain and the tile pattern onto the symmetry group quotient space, i.e. itsorbifold, O . Note that the pattern of tile “edges” on the orbifold is not necessarilyconnected and may contain simple closed curves with no natural “vertex” (i.e.no intersection with the singular locus of O ). We therefore place a single vertexat an arbitrary point on any such loop. The next step is to cut up the orbifoldalong the existing edges (coloured black) to obtain k pieces, with tile vertices,orbifold cone points, mirror boundaries and corner points marked with theirorder. Any piece that comes from a bounded tile can be treated as per the usualD-symbol process of barycentric subdivision. All other pieces require extra cutsto create a polygonal domain for barycentric subdivision. We will colour theseedges green.1. Let T be a tile from T with infinite stabiliser group H ⊂ Γ, and let
T /H be the decorated quotient space with black tile edges and vertices, conepoints, mirror boundaries and corner points marked.
T /H is a compact2-manifold with m boundary components, where these can now be mirror imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings boundaries or tile edges. Cap the boundary components to obtain a closedsurface A and compute the genus, g ( A ) ≥ α .2. Add green edges, M , to all mirror boundaries not already coloured black.If the mirror boundary has no corner points or intersection with black tileedges, then insert a vertex at an arbitrary point.3. If g ( A ) >
0, construct all possible graphs C consisting of a single vertex v and αg loops (with α = 2 if A is orientable and α = 1 if not) such that A \ C is a topological disk. We then embed the graph C in T /H , treatedas a subspace of A .4. Discard a neighbourhood of the vertex v to leave αg disjoint paths with2 αg endpoints v i .5. Construct all possible trees S on T /H that do not intersect the αg paths(except at the endpoints) such that S spans the following vertices:(a) the path endpoints v i , each of degree one.(b) cone points not already marked as belonging to a tile edge.(c) a single point on each boundary component, at an existing vertex ofminimal degree.6. Let U be the union of the green edges from M , one choice of C , and onechoice of S .7. (*) From all possibilities for U , keep those such that T /H \ U is a combi-natorial polygon with a minimal number of edges.The above algorithm is adapted from [22], but we have restricted it to insertas few vertices and edges as possible. Note that there must be at least one tileboundary component in T /H , so the construction of the tree, S , is well defined.Illustrations are provided for an example in the following section.Once we have candidate sets of green edges U , that cut each tile class intoa minimal polygon, we reassemble the fragments to form a tiling-by-disks withtwo types of edge colour. From this, we create a D-symbol for each possiblecombination of cuts found in step ( ∗ ). A canonical representative D-symbol forthe ribbon tiling will be the one that appears first in the ordering as definedin [4]. Two such D-symbols will encode equivariantly equivalent tilings if they areisomorphic in the usual sense, with the extra requirement that the isomorphismpreserves the edge-colours.The proof that this algorithm leads to a unique D-symbol encoding a ribbontiling follows from the corresponding statement for classical tilings, for which theisomorphism class of a D-symbol completely characterises the isomorphism classof the symmetry group and the tiling combinatorics. The crux of the matter isthat the above recipe leads to a unique classical tiling and D-symbol startingfrom any geometric realisation of the ribbon tiling. The constructions involvedare made in the quotient space T /H ⊂ O and so do not depend on the particularrealization of the tiling in X . Also, the edges we insert must be part of a graphthat determines a fundamental tiling of T /H . The uniqueness is guaranteed bythe fact that there are a finite number of combinatorially distinct fundamentaltilings for a given wallpaper or NEC group, and the possibility to rank D-symbols imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings by the methods of [4].The enumeration of non-fundamental tilings, in both the classical (disk-like)and ribbon tiling cases effectively constructs all the infinite-index subgroupsgenerated by a subset of the parent group generators. The subgroups are thestabiliser groups of the non-fundamental tile while the parent group generatorsare dictated by the particular fundamental domain that we start with. We ob-serve that some stabiliser subgroups are derivable from all fundamental domainsfor a particular group, but this does not always hold. In fact, there are somenon-fundamental tilings that arise from edge deletion in just one fundamentaldomain (see figure 3d for example).
5. Hyperbolic tiling examples
In this section we look at further examples of ribbon tilings that illustrate thetheory developed above and show how to apply our results in practice. We startwith an example of the classification of a ribbon tiling via coloured D-symbols.Consider the ribbon tiling in figure 5. We identify its symmetry group as G = . Figure 5b shows a fundamental domain with the cone points on itsboundary. We see that this ribbon tiling has only one class of edge and tile.The orbifold is (topologically) a sphere with four cone points, and the ribbonedge joins the cone point of order 3 to one of order 2; see figure 5c. There aretwo possible trees that span the ribbon vertex and the two other cone points,as required by the algorithm at the end of Section 4, one of these is drawn withgreen edges in figure 5c. The corresponding tilings are shown in figures 5d and 5e.The second one has only three edges bordering each fundamental tile, hence it isthe simplest classical tiling encoding this ribbon tiling. Barycentric subdivisionas shown in figure 5f leads to the coloured D-symbol shown in figure 6.Now we will show how to enumerate the ribbon tilings with symmetry group . There are nine combinatorially distinct fundamental domain tilings withthis symmetry, shown in figure 7. More information about these tilings, colouredpictures, their D-symbols and their transitivity classes can be found in [25].Classical non-fundamental tilings are built from these by deleting from the tile-boundary graph C ⊂ O , a single edge incident at a cone point with degree-1.This process results in five combinatorially distinct tilings, each with stabilisergroup either • = C or • = C .When constructing the ribbon tilings, first note the following observationsthat are a consequence of our results in the previous sections. In order to producea ribbon tile, we need to delete at least two edges from a fundamental domain.Furthermore, deleting edges that are non-adjacent in the tile boundary mustresult in a ribbon tile. If the two edges are related by a symmetry, then thissymmetry must shift all points in H by some length bounded from below, i.e.the symmetry generates a translation or glide. If the two edges are not relatedby a symmetry then the subgroup generated by the corresponding symmetryoperations must be non-abelian, and we are in the situation of Lemma 3.4. imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings (a) A tile-1-transitive rib-bon tiling with symmetrygroup . (b) The ribbon tiling from(a) with a fundamentaldomain FD of the sym-metry group. The pointsof increased symmetry onthe boundary of FD aremarked with their orders. (c) The orbifold is topo-logically a sphere with 4marked points. The greenedges are introduced toform a tree.(d) A classical tiling re-sulting from the insertionof the edges indicated infigure 5c. Each fundamen-tal tile is bordered by fouredges. (e) The simplest classi-cal tiling, with fundamen-tal tiles bordered by onlythree edges. (f) Barycentric subdivi-son of the simplest clas-sical tiling leads to thecoloured D-symbol for theribbon tiling. Fig 5: Ribbon tiling with symmetry group . imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings A(3,9) B(3,9) C(3,9)D(3,9)E(3,9)F(3,9) s s s s s s s s s Fig 6: The coloured D-symbol for the ribbon tiling from figure 5 with symmetrygroup . As in the classical setting, the D-symbol is a graph that recordsadjacencies between triangles (or chambers) using different line styles for ad-jacencies opposite a vertex (dashed), edge (dotted) and face (solid). The twonumbers associated with each chamber are face and vertex indices: ( f, v ) where( σ σ ) f ( A ) = A and ( σ σ ) v ( A ) = A are orbits in the plane X that return tothe starting chamber, A . Note that it is always the case that ( σ σ ) ( A ) = A asthere are four chambers around the midpoint of an edge. See [5, 6, 4] for furtherdetails of standard D-symbols. Our coloured D-symbol augments this graph byrecording the black or green colour for the tile edges shared by σ -adjacentchambers. imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings (a) Corresponds to thetiling QS20. (b) Corresponds to thetiling QS21. (c) Corresponds to thetiling QS22.(d) Corresponds to thetiling QS23. (e) Corresponds to thetiling QS24. (f) Corresponds to thetiling QS25.(g) Corresponds to thetiling QS26. (h) Corresponds to thetiling QS27. (i) Corresponds to thetiling QS28. Fig 7: The different fundamental tilings with symmetry group generatedby the indicated generators, with the 4-fold rotation located at vertex 4. Thenaming QS n is that used in the epinet database [25]. Further information aboutthe tilings including their D-symbols is accessible at epinet.anu.edu.au/QS n . imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings (a) Ribbon tiling withstabiliser group ∞∞ . (b) Ribbon tiling withstabiliser group ∞ . (c) Branched ribbontiling with stabilisergroup ∞ . The medialaxis is drawn in redto show the branchingstructure. Fig 8: The three distinct classes of ribbon tilings with symmetry group .For example, notice that the tiling in figure 7i can be obtained from the tilingin figure 7d by splitting the degree-4 vertex between the cone points labelled‘2’ and ‘3’ into two degree-3 vertices by introducing a new edge. Observe thatthe new edge does not intersect any cone points, that it connects two distinctvertices in the orbifold, and is not adjacent to a copy of itself. Therefore, deletingthis new edge must result in a ribbon tile with stabilizer group ∞∞ . Here, whilefigure 7i supports the ribbon tiling in figure ?? , the tiling from figure 7d doesnot.From the nine fundamental domain tilings for , just three combinatoriallydistinct ribbon tilings are possible; these are shown in figure 8. The example infigure 8a can be generated by deleting suitable subsets of edges from six funda-mental domains, namely those of figures 7c and 7e–7i. The one in figure 8b canbe built from each of the nine fundamental domain tilings with multiple distinctedge deletions from some domains giving the same tile-class. The branched rib-bon tiling in figure 8c can be found in eight of the fundamental domain tilings:the minimal triangular fundamental domain outlined in figure 7a cannot sup-port this branched ribbon. Figure 9 illustrates that this fundamental domainsupports only one type of ribbon tile, the one with stabiliser ∞ shown infigure 8b.Unlike the Euclidean ∗ example given in the previous section, all threeclasses of ribbon tilings in figure 8 can be found within a single fundamental do-main tiling. For example, the fundamental domains in figures 7c and 7f supportall three ribbon tilings.Finally, we provide a few examples of tile-2-transitive ribbon tilings. Figure10 shows two examples of tile split operations that each add a new edge to thefundamental domain of figure 7f. The tiling in figure 10c shows an example ofa non-classical tiling that with a finite stabilizer group, • = C . Deleting the imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings (a) (b) (c) Fig 9: Deleting any pair of edges from the triangular fundamental domain offigure 7a leads to a ribbon tiling with stabiliser ∞ . Each of these three tilingsis equivalent to that in figure 8b (a) Tile-2-transitive tilingwith symmetry group obtained by split-ting the tiling in figure7f. (b) A different way ofsplitting the fundamentaltile in the tiling in figure7f. (c) The ribbon tiling re-sulting from removing thegreen edges from thetiling in (b), resulting ina ribbon tiling with stabi-lizer group • . Fig 10: Tile-2-transitive fundamental tilings with symmetry group and anassociated non-fundamental tiling.green edges in this case results in a tile with the topology of an annulus.Removing different edges from these split tilings gives the examples in fig-ure 11. The tilings in figures 11a and 11b each have one fundamental tile andone non-fundamental tile with infinite stabiliser group, while 11c has two non-fundamental tiles. The ribbon tiles in 11b and 11c have stabiliser group ∞ and are non-simply connected. In fact, the fundamental group of this tile is noteven finitely generated. imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings (a) A tile-2-transitive rib-bon tiling with a non-fundamental tile obtainedby deleting the greenedges from the split tilingin figure 10a. The ribbonhas stabilizer group ∞∞ . (b) A tile-2-transitive rib-bon tiling with a non-fundamental tile obtainedby deleting an edge fromthe split tiling in figure10c. The ribbon has sta-bilizer group ∞ . (c) Gluing copies of theremaining fundamentaltile in (b) produces asecond non-fundamentaltile with stabilizer group • Fig 11: Tile-2-transitive ribbon tilings with symmetry group 2224.
6. Summary and Outlook
In this paper, we showed how to enumerate and classify periodic, locally finitetilings of the Euclidean or hyperbolic plane X , possibly with unbounded tileswith nontrivial topology. The enumeration is based on gluing together funda-mental domain tiles to create non-fundamental ones, possibly with stabilisergroups of infinite order. The method adapts well to the sequential enumera-tive aspects of the classical theory estabilished in [14] because the new glueoperations naturally extend to split fundamental domain tilings.The classification builds on the classical D-symbol by introducing colourededges that cut a ribbon or annulus tile into symmetry-related fundamentaldisks. The coloured D-symbol encoding a ribbon tiling is isomorphic to anothercoloured D-symbol if and only if the tilings may be deformed into one anotherwhile preserving their abstract symmetry. When constructing the coloured D-symbol, it is necessary to construct the possible trees on n vertices. For dis-tinguishable vertices, it is well-known that there are n n − spanning trees. Asillustrated in section 5, it is often possible to discard some trees in the con-struction of the D-symbol. However, to prevent a combinatorial explosion inthe classification algorithm, future work should focus on a priori estimates ofthe D-symbol in this enumeration and explore more efficient approaches to con-structing the coloured D-symbol of a given ribbon tiling.Note also that the theory developed here extends to the situation of hyper-bolic symmetry groups Γ, where the quotient space X / Γ has finite area but isnot compact. The situation is exactly the same as before and it makes senseto interpret the punctures that appear in the orbifold as gyration points of in-finite order (i.e. parabolic isometries of X ). For example, it would still be the imsart-generic ver. 2014/10/16 file: KolbeRobinsRibbons.tex date: April 9, 2019 olbe and Robins/Ribbon tilings case in this more general setting that deleting two non-neighbouring edges of afundamental tile-1-transitive tiling would lead to a tile with a translation in itsstabilizer subgroup, and consequently a tile with infinite area.Finally, we observe that the classification algorithm developed here could alsobe used to study graph embeddings in compact surfaces when the embedding isnot a combinatorial map.And returning to the initial inspiration of this work — the question of how toenumerate stripe patterns on the gyroid — this can now be achieved by finding(branched) ribbon tilings in the symmetry groups compatible with the coveringmap that wraps the hyperbolic plane onto this periodic surface. Acknowledgements:
The authors would like to thank Myfanwy Evans fromthe Technical University in Berlin, and Stephen Hyde and Stuart Ramsden fromthe Australian National University for fruitful discussions and enlivening talksabout the problem of classifying ribbon tilings over many years.
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