aa r X i v : . [ m a t h . D S ] M a r A NON FLC REGULAR PENTAGONAL TILING OF THEPLANE
MARIA RAMIREZ-SOLANO
Abstract
In this paper we describe the pentagonal tiling of the plane defined in the article ”A regularpentagonal tiling of the plane” by P. L. Bowers and K. Stephenson as a conformal substitutiontiling and summarize many of its properties given in the mentioned article. We show furthermorewhy such tiling is not FLC with respect to the set of conformal isomorphisms.
Introduction
The conformally regular pentagonal tiling of the plane described in the article [22]is the main character in this work. The goal is to describe this tiling as a conformalsubstitution tiling, i.e. a tiling generated by a substitution rule with complex scalingfactor γ >
1. A little theory on tilings of the plane A tile is a closed subset of the plane homeomorphic to the closed unit disk. A tiling T is a cover of the plane by tiles that intersect only on their boundaries. Moreprecisely, T is a collection of tiles, such that their union is R , and the intersectionof any two tiles t, t ′ ∈ T is the empty set or a subset of the boundary ∂t of t . Thereare three ways of constructing interesting tilings and we classify them as (1) substi-tution tilings, which are described below; (2) cut and project tilings, which invoke ahigher dimension and project globally, ie by projection of higher-dimensional struc-tures into spaces with lower dimensionality; (3) local matching rule tilings, whichare jigsaw puzzles of the plane. These classes are not disjoint, nor they are equal.For example the Penrose tiling is a substitution tiling and a cut and project tiling.See [19].Non-periodic tilings give rise to topological dynamical systems, which in turngive rise to C ∗ -algebras which in turn give rise to K -groups, which are topologicalinvariants. The construction of a topological dynamical system from a tiling is givenright after we remind the reader of the definitions of group action and topologicaldynamical systems. Supported by the Danish National Research Foundation through the Centre for Symmetry andDeformation (DNRF92), and by the Faculty of Science of the University of Copenhagen. A group action is a triple ( X, G, φ ) composed of a topological space X , an Abeliangroup G , and an action map φ : X × G → X defined by φ g : X → X , which isa homeomorphism for every g ∈ G , and φ = id and φ g ◦ φ h = φ g + h for every g, h ∈ G .A dynamical system is a group action (( X, d ) , G, φ ), where ( X, d ) is a compactmetric space called the phase space, and the group action φ is continuous. For shortwe write ( X, G ) instead of ((
X, d ) , G, φ ). The study of the topological properties ofdynamical systems is called topological dynamics, and the study of the statisticalproperties of dynamical systems is called ergodic theory. See [18].The orbit set of a tiling T is defined by O ( T ) := { T + x | x ∈ R } , where T + x := { t + x | t ∈ T } . The group R acts on the orbit set O ( T ) of a tiling T by translation, for if T ′ is in the orbit set, then so is T ′ + x for all x ∈ R .The orbit set O ( T ) is equipped with a metric d : O ( T ) × O ( T ) → [0 , ∞ [ defined by d ( T, T ′ ) < r if there is x, x ′ ∈ B /r (0) such that ( T − x ) ∩ B r (0) = ( T ′ − x ′ ) ∩ B r (0)i.e. if they agree on a ball of radius r centered at the origin up to a small wiggle.See Figure Figure 3. - - - - Figure 1.
A tiling T (Known as the chairtiling). - - - - Figure 2.
A translateof T given by T ′ := T +(2 , T of a tiling T is defined as the completion of the metricspace ( O ( T ) , d ), i.e. Ω T := O ( T ) d . The same definition of d extends to Ω T , and (Ω T , d ) is a metric space. The group R acts also on the hull by translation, for if T ′ is in Ω T then so is T ′ + x for any x ∈ R . See [15]. A patch P is a finite subset of a tiling T . A tiling satisfies the finite local complexity (FLC) if for any r > r up to a group of motion G , usually translation. The finitelocal complexity (FLC) is also called finite pattern condition . By Theorem 2.2 in[15], if a tiling T satisfies the FLC condition then the metric space (Ω T , d ) is com-pact. Hence, if a tiling T satisfies the FLC condition, then (Ω T , R ) is a topologicaldynamical system. The action φ : Ω T × R → Ω T given by φ x ( T ′ ) := T ′ + x iscontinuous by definition of the metric. NON FLC REGULAR PENTAGONAL TILING OF THE PLANE 3
Figure 3.
The two tilings T , T ′ := T + (2 ,
2) shown in Figure1 and Figure 2 agree on the yellow(large) disk of radius r , so thedistance d ( T, T ′ ) < /r . If T ′′ := T ′ + x and x is in the green(small)disk of radius 1 /r , then d ( T, T ′′ ) < /r .A topological dynamical system (Ω T , R d ) is minimal if every orbit is dense. Byconstruction, the orbit O ( T ) of T is dense in Ω T . Hence (Ω T , R d ) is topologicallytransitive. A tiling T is said to be repetitive if for every patch P ⊂ T there is an r > r contains a copy of P . A tiling T is aperiodicif T + x = T for all x = 0.By Proposition 2.4 in [15], if a tiling is aperiodic and repetitive then its hullcontains no periodic tilings.For substitution tilings, there is an alternative definition of the hull. See [14].The construction of a substitution tiling is as follows. Let P prot := { p , . . . , p N } be aset of tiles of the plane. These tiles are called prototiles. We define the substitutionrule ω with scaling factor λ > P prot by the map ω : P prot → ∪ Ni =1 O ( p i ) definedby p i ω ( p i ) := { p j + x j | p j ∈ P prot , x j ∈ R } such that ω ( p i ) tiles λp i , i.e. the tiles in ω ( p i ) overlap only on their boundaries andtheir union is λp i . Moreover ω ( p i ) is assumed to be finite. We extend the definitionof ω to translates of prototiles: Define ω : ∪ Ni =1 O ( p i ) → ∪ Ni =1 O ( p i ) by ω ( p i + x ) := ω ( p i ) + λx. Observe that ω ( p i + x ) tiles λ ( p i + x ) and not λp i + x , so all the points of the set p i + x are dilated. Thus, if ( p i + x ) ∩ ( p j + y ) = e then λ ( p i + x ) ∩ λ ( p j + y ) = λe and so ω ( p i + x ) ∩ ω ( p j + y ) tiles λe . A patch P is a finite collection of translates ofprototiles that overlap only on their boundaries. Let P patches ⊂ P ( ∪ Ni =1 O ( p i )) bethe collection of patches derived from the prototiles. We now extend the definitionof ω to patches. Let ω : P patches → P patches be defined by P ω ( P ) := [ t ∈ P ω ( t ) . MARIA RAMIREZ-SOLANO
Observe that ω ( P ) tiles the set λ | P | , where | P | is the union of all the tiles in P .We define the N × N substitution matrix for the substitution map ω with prototiles P prot by A := ( a ij ) where a ij is the number of copies of prototile p i in ω ( p j ).It turns out that ω n is also a substitution map with scaling factor λ n . Moreover,the substitution matrix of ω n is A n . We say that the substitution map ω is primitiveif its substitution matrix A is primitive.If the substitution matrix A is primitive , then there is k > A k ) ij > p i appears inside the patch ω k ( p j ) for any i, j .If the substitution ω is primitive, we have p j + x ∈ ω k ( p j + y ) for some x, y ∈ R .We can then construct an increasing sequence of supertiles p j + x ⊂ ω k ( p j + x ) ⊂ ω k ( p j + x ) ⊂ · · · ω nk ( p j + x n ) ⊂ · · · , for some x , x , x , · · · . Their union might not necessarily cover the entire plane,as λ nk p j + λ nk x n might only cover a section of the plane. The substitution is saidto force its border if there is a m ≥ j such that ω m ( p j ) knows itsneighbor tiles. Any substitution can be turned into a substitution that forces itsborder, simply by introducing collared tiles, which are tiles that remember theirneighbors and their location relative to the tile. If the substitution forces its border,then the union of the above increasing sequence will cover the plane, and thus wehave created a tiling of the plane. If the substitution does not force its border, thenwe use collared tiles instead together with the new substitution.Define Ω to be the set of tilings T whose tiles are translations of the prototilesand each patch P in T is contained in a supertile ω n ( p i + x ) for some n, i, x . If T is a tiling in Ω then so is ω ( T ) := [ t ∈ T ω ( t ) . We now extend ω to tilings. Define ω : Ω → Ω by T ω ( T ).The space Ω is said to satisfy the finite local complexity (FLC) if for each r > P ⊂ T ∈ Ω of diameter less than r up totranslation. Hence if Ω is FLC then every tiling in it is FLC. Moreover if T is FLCthen Ω T is FLC. If the substitution map ω is primitive and injective, and the spaceΩ satisfies the FLC condition then: the space Ω is nonempty by Proposition 2.1 in[14]. The map ω is surjective by Proposition 2.2 in [14]. The space Ω contains noperiodic tilings by Proposition 2.3 in [14].The metric d defined before defines a metric on Ω. Under the same conditions that ω is primitive and injective, and Ω satisfies the FLC condition, the group action((Ω , d ) , R ), where R acts by translation, is a topological dynamical system, andby Corollary 3.5 in [14] it is minimal. Hence for any T ∈ Ω we haveΩ = O ( T ) d = Ω T . Moreover, by Proposition 3.1 in [14], the substitution map ω : (Ω , d ) → (Ω , d ) is atopologically mixing homeomorphism. In particular ω is continuous. Let T be inΩ. Let Ω ′ be the set of tilings T ′ such that T ′ is a tiling whose tiles are translatesof the prototiles P prot and T ′ is locally isomorphic to T . By Corollary 3.6 in [14]we have Ω = Ω ′ .Also, every primitive substitution tiling is repetitive. By Theorem 1.1 in [21], fora substitution tiling T that satisfies the FLC condition, the substitution map ω isinjective if and only if its hull Ω T contains no periodic tilings.In conclusion we have, NON FLC REGULAR PENTAGONAL TILING OF THE PLANE 5 T FLC = ⇒ Ω T compact. T FLC: ω injective ⇐⇒ Ω T has no periodic tilings. T FLC, substitution tiling, ω primitive, = ⇒ T repetitive, FLC ⇐⇒ (Ω T , R d ) minimal. T substitution tiling, ω primitive, = ⇒ T repetitive,aperiodic = ⇒ Ω T has no periodic tilings.Ω FLC, ω primitive, injective = ⇒ Ω = ∅ , surjective, has no periodic tilings.Ω FLC, ω primitive, injective = ⇒ (Ω , R ) minimal. Hence Ω = Ω T .. Thus if T is FLC, aperiodic and ω is primitive, then (Ω T , R ) is a minimal topo-logical dynamical system; Ω T has no periodic tilings, and ω is injective.For each prototile p i choose a point x i ∈ ( p i ) ◦ in the interior of it. We say that x i is a puncture of p i . The puncture of any translate t = p i + x is defined by x ( t ) := x i + x . Define the discrete hullΞ := { T ∈ Ω | T has a puncture at the origin } . Since the punctures do not lie on the boundaries, the choice of p i − x i is unique.If T is FLC, then Ξ is a Cantor set: compact, totally disconnected, and no isolatedpoints. A basis for Ξ is given by the cylinder sets U ( P, t ) defined as follows: Let P be a patch of T and t a tile in P . Define U ( P, t ) := { T ′ ∈ Ω T | P − x ( t ) ⊂ T } tobe the set of tilings that contain P − x ( t ). That is, U ( P, t ) is a subset of Ξ suchthat the patch P − x ( t ) is centered at the origin and anything can be outside thepatch. Then U ( P, t ) is clopen in the relative topology of Ξ, and such sets generatethe relative topology of Ξ. See [15].The action of translation induces an equivalence relation R on Ω by declaring twotilings T, T ′ ∈ Ω to be equivalent if they are translates of each other i.e. if T = T ′ + x for some x ∈ R . Let R ′ be the restriction of R to Ξ, which is an equivalence relationon Ξ. The set Ξ is a full transversal of Ω with respect to R because [ T ] R ∩ Ξ = ∅ iscountable for any tiling T ∈ Ω. Observe that the natural map Ξ → Ω /R given by T [ T ] R is surjective but not injective.By [15], we can construct the C ∗ -algebras C ∗ ( R ′ ), C ∗ ( R ). These C ∗ -algebras arestrongly Morita equivalent because Ξ is a transversal to Ω relative to R , and R ′ isthe restriction of R to Ξ, and because Ξ satisfies the following three conditions: (1) T ′ ∈ Ω = ⇒ T ′ + x ∈ Ξ for some x ∈ R . (2) T ′ ∈ Ξ = ⇒ { T ′ + x | < | x | < ε } = ∅ for some ε >
0. (3) Ξ is closed in Ω.The pair ((Ω , d ) , ω ) is also a dynamical system of its own. The group action is Z ,where the homeomorphisms are ω a , a ∈ Z . It is a Smale space.
2. Combinatorial Tilings An n -cell on a topological space X is a subspace homeomorphic to the open n -disk D n := { v ∈ R n | || v || < } . Hence cells are open in X . A cell decomposition of atopological space X is a partition of the space into n -cells. Definition . A pair ( X, E ) consisting of a Hausdorff space X and a cell decomposition E of X is called a CW-complex if the following threeconditions are satisfied:(1) For each n -cell e ∈ E , there exists a continuous map Φ e : D n → X fromthe closed unit n -disk D n to X that takes the boundary S n − into the n − X n − , and the restriction of this map to the interior D n isa homeomorphism.(2) For any cell e ∈ E the closure e intersects only a finite number of othercells.(3) A subset A ⊂ X is closed if and only if A ∩ e is closed in X for each e ∈ E .A nice result from CW-complexes which will be used extensively is the following:If ( X, E ) is a CW-complex and f : X → Y is a map between two topological spaces,then f is continuous if and only if f restricted to each n -cell is continuous if and MARIA RAMIREZ-SOLANO only if f restricted to each n -skeleton X n is continuous. Moreover, X n is obtainedfrom X n − by gluing the n -cells in X .The dimension of a CW-complex is said to be n if all the cells are of degree atmost n . A combinatorial tiling is a 2-dimensional CW-complex ( X, E ) such that X is homeomorphic to the open unit disk D := D . The combinatorial tiles are theclosed 2-cells. We also say that a face is a closed 2-cell, an edge a closed 1-cell, anda vertex a 0-cell. Example . If T is a tiling of the plane, then T has the structure of a 2-dimensional CW-complex, where the 2-cells are the interior of the tiles, the 1-cellsare the interior of the edges of the tiles, and the 0-cells are the vertices of the edgesof the tiles. Hence, under this identification, ( C , T ) is a combinatorial tiling.By the cellular approximation theorem, a continuous map between CW-complexescan be taken to be cellular, i.e. that it maps the n -skeleton of the domain to the n -skeleton of the range. However, the maps between CW-complexes that we willconsider in this work are stronger than cellular maps, for they map cells to cells.We call such maps cell-preserving maps.We borrow the definition of subdivision of a combinatorial tiling from [3]. Definition . Let ( X, E ) and ( X, E ′ )be two combinatorial tilings with same topological space X . We say that ( X, E ′ ) isa subdivision of ( X, E ) if for each cell e ′ ∈ E ′ , there is a cell e ∈ E such that e ′ ⊂ e . Figure 4.
No regu-lar pentagonal tiling onthe plane.
Figure 5.
A pentago-nal tiling.
3. Conformal regular pentagonal tiling of the plane T Figure 4 shows that we cannot tile the plane with regular pentagons. However, wecould deform the pentagons and obtain a pentagonal tiling like the one in Figure5. Although this tiling has many nice properties, we can hope for more. Wecan construct a tiling with the same combinatorics, but where the tiles are socalled conformally regular pentagons, and the tiling looks like the one in Figure6. The article ”A regular pentagonal tiling of the plane” by Philip L. Bowers andKenneth Stephenson in [22] gives a construction of this tiling using the theory ofcircle packings on the above combinatorics. They use circle packings to impose anatural geometry on the above combinatorics. This gives the nice feature to the
NON FLC REGULAR PENTAGONAL TILING OF THE PLANE 7
Figure 6.
The tiling T , which is a conformal regular pentagonaltiling of the plane.tiles of being almost round. We give a brief summary of the construction andextract/deduce on our own some of the remarkable combinatorial and geometricproperties of the tiling. Before we proceed, we need to introduce some terminologyand results from Riemann surfaces.
4. Riemann Surfaces A Riemann surface is a connected Hausdorff topological space S together with ananalytic atlas, that is, a Riemann surface is just a one dimensional complex mani-fold. See pages 33, 39 in [1]. A map f : S → S ′ between two Riemann surfaces issaid to be analytic if for every chart φ U : U → C of S and every chart φ V : V → C of S ′ , the map φ V ◦ f ◦ φ − U is analytic. We should note that a chart φ U is a homeomor-phism onto its image. Two Riemann surfaces are said to be conformally equivalent if there is a bijective analytic map between them (hence an analytic homeomor-phism). A connected Hausdorff topological space S with an analytic atlas is alwaysequivalent to the same space S with a different analytic atlas, since the identitymap gives a conformally equivalence between them. Hence if we treat conformallyequivalent Riemann surfaces as identical, we do not need to worry about whichatlas we equip S with. A topological closed disk is a topological space homeomor-phic to the closed unit disk. In particular it is compact. For example a regularEuclidean pentagon is a topological closed disk. A conformally regular pentagon isa Riemann surface P with boundary and five distinguished points on its boundary,which we call corners, such that: (1) it is a topological closed disk, (2) the interior P ◦ is conformally equivalent to the interior of a regular Euclidean pentagon P ′◦ ,(3) the bijective analytic map from P ◦ to P ′◦ extends continuously to the bound-ary, mapping corners to corners. We should emphasize that conformally regularpentagons are analytic on their interior, extend continuously to the boundary andthey are not conformal on the boundary, so the angles on the boundaries are notpreserved. An example of this is of course a regular Euclidean pentagon. Somemore exotic examples of conformally regular pentagons are shown in Figure 7, and MARIA RAMIREZ-SOLANO
Figure 7.
The prototiles of T . The interior angles are either π/ π/ π/ conformally regular pentagons of the plane .We say that two conformally regular pentagons of the plane are extended confor-mally equivalent if they extend to open sets which are conformally equivalent. Twopentagons from Figure 7 cannot be extended analytically beyond their boundariessince their interior angles do not match. Hence these are not extended conformallyequivalent.
5. Conformal substitution
We name T the tiling shown in Figure 6, which is also known as a conformalregular pentagonal tiling of the plane. The reason is that all its tiles are extendedconformally equivalent to the pentagons shown in Figure 7, and so the boundaryangles are preserved. This tiling is a conformal substitution tiling. That is, thereis a substitution map ω with complex dilation λ ∈ C that replaces a tile t with apatch ω ( t ) satisfying: (1) the tiles in ω ( t ) are conformal regular pentagons whichare extended conformally equivalent to the prototiles; (2) the union of all the tilesof ω ( t ) is λt . The prototiles are the three pentagons shown in Figure 7. Thesubstitution map is shown in Figure 8, where the complex dilation constant is λ = ( − / ≈ . e iπ/ - a dilation by 3.2 and a rotation by π/ λt with conformal copies of the prototiles, and notwith traslations/rotations of the prototiles, is a sign of a generalization of thestandard theory for substitution tilings. The construction in [22] of the tiling T isas follows.
6. Construction of the conformal regular pentagonal tiling T . The story of the construction of T involves five steps. The first one is to constructa combinatorial tiling K , which contains the combinatorics of T . We then equip itwith a piecewise affine structure using regular pentagons, and then with a confor-mal structure to obtain a simply connected non compact Riemann surface. Finally,using a generalization of the Riemann mapping theorem, we map this one dimen-sional complex manifold onto the plane to obtain our tiling T . However, we needcircle packing theory to see a drawing of the tiling T . If no confusion arises we will NON FLC REGULAR PENTAGONAL TILING OF THE PLANE 9
Figure 8.
The substitution map for the tiling T .use the same symbol K to denote K with any of the structures, but if we want toemphasize which structure we are referring to then we will write K , K aff , K conf todenote that K is equipped with the combinatorial, piecewise-affine, and conformalstructure, respectively. Recall that a combinatorial tiling is a CW-complex home-omorphic to the open unit disk, and so it is a pair consisting of a topological spaceand a partition. Usually, no ambiguity arises by denoting the topological space andthe partition with the same symbol, for if we talk about cells, we talk about thepartition, and if we talk about points, we talk about the space.The construction of the combinatorial tiling K is the following. Start with a combinatorial pentagon K , which is a closed topological disk with five distinguishedpoints on its boundary. We write combinatorial to emphasize that at this stage weonly care about combinatorics. That is we can think of it abstractly as one face withfive edges and five vertices. Using the combinatorial subdivision rule from Figure9, we subdivide K into six combinatorial pentagons. The result is a combinatorialflower K , which is shown in Figure 10. We identify K with the central pentagonof K , and we write K ⊂ K . Repeat this subdivision for the new six pentagonsto obtain the combinatorial superflower K shown in Figure 10. We identify K with the central flower of K and so we have K ⊂ K ⊂ K . Repeating thissubdivision n times we obtain an increasing sequence of combinatorial superflowers K ⊂ K ⊂ · · · K n . The union of all K n is a combinatorial tiling whose tiles arecombinatorial pentagons, each of them attached as in Figure 6. An alternativeconstruction of K n from K n − is using a so called reflection rule, which is shown inFigure 11. This rule consists in (1) reflecting the central pentagon K across eachof its five edges; then (2) the two edges coming out from the central pentagon areidentified with each other, and so 5 identifications are made, each corresponding toa corner of the central pentagon, as seen in Figure 11. The result is K . Repeat thesame reflection procedure of the superpentagon K across each of its five superedgesand glue them as before. The result is K , and so on: Reflecting K n − across eachof its superedges and glue them as in Figure 11 we get K n .In summary, the combinatorial tiling K is a CW-complex whose cells are shownin Figure 6 and whose topological space is (homeomorphic to) C . It has a centralpentagon K , and its group of combinatorial automorphisms (i.e. cell-preservinghomeomorphisms) Aut ( K ) is the dihedral group D , which is composed of five Figure 9.
Subdivision map.
Figure 10.
The central pentagon K , combinatorial flower K ,and superflower K . Figure 11.
Reflection rule.rotations with respect to K , and five reflections with respect to K and a vertex of K . Another important property of the combinatorial tiling K is that its verticeshave either degree 3 or 4; that is, each vertex of K is a vertex of either 3 faces or4 faces.We now equip K with a piecewise affine structure: Equip the one skeleton K with the unit edge metric, making each edge isometric to the unit interval. Thenextend this metric to faces so that each face is isometric to a regular pentagon ofside-length 1. The distance between two points is defined to be the length of theshortest path between them. This amounts to replacing each pentagon of K withEuclidean regular pentagons of side-length 1, all glued along their edges accordingto the combinatorics of K . Observe that on each pentagon we use the Euclideanmetric and that the resulting metric ensures compatibility with the combinatorialstructure of K ; for example, the isometric cell-preserving automorphisms of K arestill the five rotations and reflections of the dihedral group D .As a side note, we wondered how the piecewise affine space K aff looked like.The pentagonal flower K is simply a half-dodecahedron. The superflower K should look like Figure 12, but unfortunately, Mathematica shows otherwise. Thethree degree vertices and flatness of the pentagons, force us to glue a face of 5dodecahedrons around 5 faces of a dodecahedron, which is not possible - there are NON FLC REGULAR PENTAGONAL TILING OF THE PLANE 11 gaps in between. The dihedral angle of a dodecahedron is arccos − √ , while theangle to be filled is of π − arccos , which leaves a gap of about 10 degrees. Figure 12. K as agluing of six half dodec-ahedrons. Figure 13. K is notthe gluing of six halfdodecahedrons.We now equip K aff with a conformal structure: The piecewise affine space K aff is a nonflat surface. The corners and edges will be smoothed by charting theminto the plane. For each edge and each vertex we define a chart: Suppose that e is an oriented edge of K aff with pentagon P ∈ K aff on its left and pentagon P ∈ K aff on its right. See Figure 14. Define the open set U e := ( P ∪ P ) ◦ asthe interior of the union of the pentagons. We now introduce a coordinate systemon U e by identifying e with the interval ] − / , /
2[ and P and P with the tworegular pentagons Q , Q that have in common this interval with Q being abovethe x -axis and Q below. We define V := ( Q ∪ Q ) ◦ ⊂ C as the interior of theunion of these two pentagons; see Figure 14. The identification map φ e : U e → V is given by φ e ( z ) := z . The map φ e is the chart containing edge e , it is indexed byit, and it is an isometry. The intersection of two domains U e , U e ′ is (1) is itself if e and e ′ are the same as nonoriented edges; (2) is a pentagon P if e and e ′ are twoedges of P ; (3) is the empty set otherwise. For the first case, the transition map is φ e ◦ φ − e ′ ( z ) = − z if e and e ′ have opposite orientation, else φ e ◦ φ − e ′ ( z ) = z . Forthe second case, the transition map is φ e ◦ φ − e ′ ( z ) = ( z − z ) e m πi/ + z , where z = ± i (1 + √
5) is the baricenter of the pentagon, m = 0 , , , , z of the pentagon) the edge e ′ tomatch edge e , but this is assuming that both edges have same orientation; if theyhave opposite orientations then φ e ◦ φ − e ′ = φ e ◦ φ − − e ′ ( − z ), where − e ′ is the edge inthe opposite direction of e ′ . See Figure 15. Hence, the transition maps for chartson edges are all analytic.If v is a d -degree vertex of K aff and U v is the open metric ball of radius 1 / v , and V ′ ⊂ R is the open ball of the plane of radius 1 / φ v : U v → V ′ is defined by φ v ( z ) = z / (3 d ) , where z := re iθ , and 0 ≤ θ ≤ π d and 0 ≤ r < and the degree of the vertex is d = 3 ,
4. See Figure 16 and Figure 17. Observe that φ v ( re d π i ) = r / (3 d ) e d π i d = r / (3 d ) e πi = r / (3 d ) = φ v ( r ), as expected. Notice that we could as well have usedopen balls of radius 1 / / Figure 14.
The chart φ e : U e → V , where U e = ( P ∪ P ) ◦ . Figure 15.
The tran-sition map φ e φ − e ′ ( z ) =( z − z ) e πi / + z ,where z = i (1+ √ / V ′ is (1 / / (3 d ) which takes the values 0 . . Figure 16.
The chart φ v : U v → V ′ . Figure 17.
The flat-ten version of the chart φ v : U v → V ′ when d =3.The intersection of the domains U v , U v ′ for the charts φ v , φ v ′ for distinct verticesis always empty. However, the intersection of the domains U e , U v is nonemptywhenever v is a vertex of the edge e . For such case, the transition maps φ e ◦ φ − v : φ v ( U e ∩ U v ) → φ e ( U e ∩ U v ), φ v ◦ φ − e : φ e ( U e ∩ U v ) → φ v ( U e ∩ U v ) are alsoanalytic. For example, if the ” x -axis” on the ball U v matches the edge e then φ v ◦ φ − e ( z ) = ( z − z e ) (10 / d ) and φ e ◦ φ − v = ( φ v ◦ φ − e ) − , where z e := φ e ◦ φ − v (0).See Figure 18. Since the transition maps are analytic maps, we have constructedan atlas - our conformal structure, and thus we have turned K aff into a simplyconnected noncompact Riemann surface K conf . This structure also preserves thecombinatorial structure. For instance, the analytic cell-preserving homeomorphismson K conf are still the five rotations and five reflections of the dihedral group D .The creation of our tiling is obtained as follows. A generalized version of theRiemann mapping theorem (called the uniformization theorem) tells us that there NON FLC REGULAR PENTAGONAL TILING OF THE PLANE 13
Figure 18.
A transition map φ v ◦ φ − e ( z ) = ( z − z e ) / (3 d ) with d = 3.is an essentially unique analytic map Φ that sends our Riemann surface bijectivelyonto either the Euclidean plane or hyperbolic disk. We define our conformallyregular pentagonal tiling of the plane T := { Φ( f ) | f ∈ K } , and so ( C , T ) or ( D , T )is a CW-complex, and Φ becomes a cell-preserving map. By Schwarz reflectionprinciple, the subdivision map (or more precisely the reflection map) ω : K → K induces an analytic cell-preserving homeomorphism ω : K conf → K conf , and thus α := Φ ◦ ω ◦ Φ − is a cell-preserving analytic homeomorphism. The fact that ω ( K ) = K (i.e. the self similarity of K ) implies that α increases area. Sincenone of the automorphisms on the hyperbolic disk increase area, α must be anautomorphism of C . Hence the image of Φ is the Euclidean plane C .The map Φ : K conf → C tells us how to flatten our Riemann surface K conf onto the Euclidean plane. Unfortunately, the Riemann mapping theorem (and itsgeneralized version) is an existence theorem. It does not tell us how to draw it.We use the theory of circle packings to do this, which is explained in full detailin [23]. First we triangulate our space K aff by triangulating each pentagon asfollows: Put a vertex on the center of the pentagon and join this vertex with everycorner of the pentagon. The resulting triangulation of K aff is denoted by K ′ . Theanalytic atlas of K conf is also an analytic atlas of K ′ and so K ′ is a Riemannsurface which is conformally equivalent to K conf . Circle packing theory gives usan essentially unique collection of Euclidean circles P K ′ whose centers in C areindexed by the vertices of this triangulation. These circles will not overlap, andwill follow the combinatorics of K ′ in the sense that every vertex corresponds toa circle, every edge corresponds to two mutually tangent circles, and every face(i.e. triangle) corresponds to three mutually tangent circles, all respecting theorientation. The carrier Carr ( P K ′ ) ⊂ C is defined as the CW-complex whose2-cells are the Euclidean triangles in C induced by the faces of K ′ . The carrieris a piecewise affine space. Since it has the same combinatorics of K ′ , we candefine a piecewise affine cell-preserving map Φ : K ′ → Carr ( P K ′ ). The uniformlower bounds on the triangles in Carr ( P K ′ ) and K ′ implies that Φ is a k -quasiconformal map for some k >
1. Recall that k -quasiconformal (or k -qc) maps aregeneralizations of analytic maps. A 1-qc map is actually an analytic map, and composition of a k ′ -qc map and a k ′′ -qc map is a k ′ k ′′ -qc map, for k ′ , k ′′ ≥ ◦ Φ − : C → Carr ( P K ′ ) is a k -qc map. By Liouville’s theorem, there isno quasiconformal mapping of C onto a proper subset of C , and so Carr ( P K ′ ) = C .We now refine the affine space K ′ into the triangulation K ′′ by introducing a vertexat the midpoint of each edge of K ′ and then joining these vertices together with newedges. We say that K ′′ is a hexagonal refinement of K ′ . By the same procedure weobtain a bijective k -qc map Φ : K ′′ → C , and by induction we obtain a sequenceof bijective k n -qc maps Φ n : K ( n ) → C . The maps Φ n ◦ Φ − are also known asdiscrete conformal mappings. Since we are doing hexagonal refinement, all k n arethe same as k , and therefore Φ n ◦ Φ − converges to some k -qc map Ψ, (after somenormalization). By the Rodin-Sullivan Theorem (Theorem 19.1 in [23], also knownas Thurston’s conjecture) Ψ restricted to any triangle of K ′ is analytic and thus,Ψ is analytic everywhere except on a set of measure zero, which implies that Ψ isanalytic everywhere.Define T n := { Φ n ( K aff ) | f ∈ K aff } . We say that T n is an approximation of T . The construction of the first approximation T of T (i.e. n = 1) is shown inFigure 19 and is done as follows: each Carr ( P K ′ i ) is normalized so that their centralpentagons agree, where K ′ i is the triangulation of K i ⊂ K ; their union converge tothe space Carr ( P K ′ ). - - Figure 19.
The ap-proximation T of T . red Figure 20.
Thetilings T (the blue one)and λT (the red one)where λ = | λ | e ( π/ i , | λ | = 18 / .The properties of our tiling T are listed in the following Proposition. Recall thatthe automorphism of C are the complex linear maps. Two subsets of the plane aresaid to be euclideanly similar if there is a conformal automorphism of C that mapsone subset to the other. Thus, euclideanly similarity filters out location, scale androtational effects from the subsets when comparing them. Proposition . Our conformal regular pentagonal tiling T has the followingproperties. (1) Each tile of T is extended conformally equivalent to one of the three pen-tagons shown in Figure 7. We call these three pentagons, the prototiles of T . (2) With a single tile τ ∈ T , we can reconstruct the entire tiling T . NON FLC REGULAR PENTAGONAL TILING OF THE PLANE 15
Figure 21.
Prototiles of K .(3) Each tile τ ∈ T is euclideanly similar to at most ten tiles in T . (4) The tiling T is aperiodic. (5) The support of the 1-skeleton ( λT ) is a subset of the support of the one-skeleton T , where λ = ( − / = 18 / e iπ/ ≈ . e iπ/ . See Figure20. (6) From the circle packing approximation T of T we can obtain good approx-imations of the tiles of T . (7) There are infinitely many tiles of diameter less than n for some n > . (8) The set of diameters of the tiles of T is an unbounded set. Proof. (1). There are three prototiles of K conf , namely those shown in Figure21. Since Φ : ( K, atlas ) → C is a conformal map, their images are the prototilesof T . What remains to be shown are the interior angles of such prototiles. If thedegree of a vertex v ∈ K is d , then we have d pentagons glued at v . The interiorangles of these pentagons at v are 3 π/
5. Recall that the angle between two smoothcurves γ , γ which cross at v on a Riemann surface is defined as the angle betweenthe charted curves φ v ◦ γ , φ v ◦ γ at φ v ( v ) = 0. Thus, the interior angles of thesepentagons at v charted by φ v are (3 π/ / (3 d )) = (2 π/d ). Since d = 3 ,
4, theinterior angles of these charted pentagons is either 2 π/ π/ π/
2. SinceΦ : (
K, atlas ) → C is a conformal map, the interior angles of the pentagons arepreserved.(2). This is shown on page 9 of the article [22]. The idea is that each edge e ofthe tile τ is an analytic arc, and we can use the corresponding local (anticonformal)reflection to map τ to the neighboring tile across e . By repeated reflections in edges,we reconstruct T = T ( τ ).(3). Since the automorphisms of the combinatorial tiling K is the dihedral group D , the automorphisms of the tiling T are the Φ-conjugations, that is, if α ∈ Aut ( K )then Φ ◦ α ◦ Φ − : C → C is a (conformal) automorphism of T ; in particular, it isa conformal automorphism of C and hence a M¨obius transformation. Thus, T hasten automorphisms. Since M¨obius transformations of C are exactly combinationsof rotations dilations and translations, τ and Φ ◦ α ◦ Φ − ( τ ) are similar. If twotiles τ := Φ( f ), τ := Φ( f ) are similar in C , then by (2), the tiling T ( τ ) issimilar to the tiling T ( τ ), hence combinatorially equivalent. Thus, there exists acombinatorial automorphism of K mapping f to f . By page 9 in the article [22],the converse is also true if we use their normalization, which places the center ofthe tile corresponding to K at the origin. In summary, two tiles are similar if andonly if there is an automorphism of T that maps one tile to the other. Thus, thereare at most 10 similar tiles to τ (five reflections and five rotations with respect tothe central pentagon).(4) If T was periodic, then there would be an x ∈ R such that T and T + x aretranslates, i.e T = T + x . Hence T = T + x = ( T + x ) + x = T + 2 x = . . . = T + nx , n ∈ N . Since the automorphisms of T are in total ten, a finite number, and atranslation is an automorphism, T cannot be periodic. (5),(6). In [22], it is shown that the central tile of T can be subdivided withthe subdivision rule in a conformal way, and using Schwarz reflection principle, wecan divide the whole T . The result is a tiling T ′ whose combinatorics are ω ( K ),and an analytic bijective map α : T → T ′ mapping each tile to its subdivision.Hence ω = Φ − ◦ α ◦ Φ, and thus ω is analytic. In page 9 of the same article, itis shown that the map α : C → C is given by α ( z ) := λz , λ = ( − / (undersome normalization settings). Hence (5). We show (6) only when the tile is thecentral pentagon, since we have notation for this, and the same argument appliesfor any other tile of T . Let τ n := Φ( K n ). Since K n is the subdivision of K n − , τ n = α ( τ n − ) = λτ n − = · · · = λ n τ . Hence τ n and τ are euclideanly similar.Recall that T := Φ ( K ) is the first approximation of T . Let t n := Φ ( K n ) be thefirst approximation of the supertile τ n . Since the interior angles of the boundary of t n are closer to 2 π/ n gets, t n gives a better approximation of τ than t . See Figure 22.(7) In the proof of Proposition 5.1 of the article, it is shown that for each tile τ there is a sequence ( τ nj ) n ∈ N of tiles that converge in Hausdorff metric to b ¯ τ , where b ≈ .
3. See Figure 23 and Figure 24. Recall that the Hausdorff metric is definedon compact sets
A, B by d ( A, B ) = max(max x ∈ A d ′ ( x, B ) , max y ∈ B d ′ ( A, y )), where d ′ ( x, B ) is the smallest distance between x and B . See Figure 25. Figure 22.
The ap-proximations t i , i =0 , , , τ . Figure 23.
The col-ored pentagons, oncenormalized as in Figure24, converge in Haus-dorff measure to b τ ,where τ is the centralpentagon.(8) Let τ ∈ T be a tile. By the proof of (7), there is a tile τ satisfying b ′ diam ( τ ) < diam ( τ ), where b ′ < b ≈ .
3, say b ′ := 1 .
1. Repeating the same ar-gument for τ , we find a tile τ satisfying b ′ diam ( τ ) < b ′ diam ( τ ) < diam ( τ ). Re-peating the same argument n times we obtain a tile τ n ∈ T satisfying b ′ n diam ( τ ) Let G be the group of isometries. Suppose that T has FLC with respectto G . Let τ be the central tile of T . By part (7) of the previous Proposition 6.1, NON FLC REGULAR PENTAGONAL TILING OF THE PLANE 17 Figure 24. The colored pentagons (rotated and centered) fromFigure 23 converge in Hausdorff measure to b τ . Figure 25. Define a := max x ∈ A d ′ ( x, B ), b := max y ∈ B d ′ ( A, y )).The Hausdorff distance between A and B is max ( a, b ) = a .there is a sequence of tiles of diameter less than b · diam ( τ ), b ≈ . 3. Since T hasFLC with respect to G , there is a finite number of patterns; that is, each of thesetiles are euclideanly similar to a finite number of them. This is a contradiction,because by part (3) of the previous Proposition 6.1 each tile is euclideanly similarto at most 10 other tiles.Informally, FLC says that if we consider all regions of a fixed size, up to somenotion of equivalence, there are only finitely many. We don’t need a group; we neednotion of equivalence. In the classical theory, everything is sitting in the plane andit is natural to use translations or isometries. If the notion of equivalence we areusing does not preserve the notion of size, it is useless, for what is the point oftaking two regions of size r and asking if there is an isomorphism between themthat doubles the size? This isn’t the end of the problems because we will losecompactness and so on as well. But it is enough to give up on the idea. Take forexample the sequence of tiles of diameter less than b · diam ( τ ), b ≈ . . 3, but the patches around them are converging combinatorially to K . What doesthis sequence converge to? We don’t know.Thus we have to abandon the notion of FLC with respect to the set of conformalisomorphisms that are defined between open subsets of the plane. However, we willshow in another article that the combinatorial tiling K has FLC with respect tothe set of isomorphisms that are defined between subcomplexes of K . Acknowledgments The results of this paper were obtained during my Ph.D. studies at University ofCopenhagen. I would like to express deep gratitude to my supervisor Erik Chris-tensen and to Ian F. Putnam, Kenneth Stephenson, Philip L. Bowers, Bill Floyd,whose guidance and support were crucial for the successful completion of this work. References 1. A. F. Beardon, A Primer on Riemann Surfaces. Cambridge University Press, 1984.2. Jean Bellissard, Riccardo Benedetti, and Jean-Marc Gambaudo, Spaces of Tilings, FiniteTelescopic Approximations and Gap-Labeling. Commun. Math. Phys. 261, (2006) 1-41.3. J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules. 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