SUBSTITUTION TILINGS WITH DENSE TILE ORIENTATIONS AND n FOLD ROTATIONAL SYMMETRY

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1 SUBSTITUTION TILINGS WITH DENSE TILE ORIENTATIONS AND FOLD ROTATIONAL SYMMETRY D. FRETTLÖH, A.L.D. SAY-AWEN, AND M.L.A.N. DE LAS PEÑAS Abstract. It is show that there are primitive substitutio tiligs with dese tile orietatios ivariat uder -fold rotatio for {, 3, 4, 5, 6, 8}. The proof for dese tile orietatios uses a geeral result about irratioality of agles i certai parallelograms.. Itroductio From the discovery of Perose tiligs i the 70s [3] ad of quasicrystals i the 80s [9] evolved a theory of aperiodic order. Oe mai method to produce iterestig patters showig aperiodic order is a tile substitutio. For a more precise descriptio see below. The idea is illustrated i Figure : a tile substitutio is a rule of how to elarge a give prototile (or a set of several prototiles) ad dissect it ito cogruet copies of the prototiles. The rule ca be iterated to fill larger ad larger regios of the plae. Formally oe cosiders a fixed poit of the substitutio: a Figure. Two iteratios of the substitutio for the piwheel tilig. ifiite tilig T of the plae ivariat uder the substitutio rule. This fixed poit T yields the hull of the tilig: the closure of the image GT of T, where G is a group actig o R (usually all traslatios i R, or all rigid motios), ad closure is take with respect to the local topology. For details see below, for more details see for istace [3]. Several mathematical fields iteract i the theory of aperiodic order. A lot of literature is dedicated to studyig the topology of the hull of a aperiodic tilig. Oe way to do this is to compute its cohomology groups. For substitutio tiligs this ca be doe by the methods itroduced i []. For the piwheel tilig this was doe i [4] ad [9]. Oe problem for the piwheel tilig is that the tiles occur i ifiitely may differet orietatios. More precisely: the piwheel tilig has dese tile orietatios (DTO), i.e. the orietatios of the tiles are dese o the circle. For the treatmet of hulls of tiligs with DTO i the cotext of dyamical systems see [8]. A further problem is that the hull of the piwheel tilig cotais six differet tiligs ivariat uder -fold rotatio. These tiligs correspod to coe sigularities of the quotiet of the hull by the circle. These give rise to a torsio part i the secod cohomology group H of the hull. I particular, m tiligs i the hull that are ivariat uder -fold rotatio cotribute a Z m subgroup to H, where Z deotes the cyclic group of order [4, Theorem ]. I view of this problem Jea Saviie [6] asked i 03 for which values of primitive substitutio tiligs with Date: February, 06.

2 BIELEFELD, ATENEO, AND ATENEO DTO ca be ivariat uder -fold rotatio. (For the defiitio of primitivity see below.) This questio motivated this paper. Our mai result is the followig. Theorem. There are primitive substitutio tiligs with DTO that are ivariat uder rotatio by π for {, 3, 4, 5, 6, 8}. These tiligs are ot mirror symmetric, hece they occur i pairs for each such. Hece we have m i the discussio above, ad the cotributios Z m are ot trivial. It is likely that the idea carries over to ay N, but sice the proof is costructive (ad the costructios become tedious for large ) we ca deal oly with the small cases here. The cases {3, 4, 6} are cosidered i more detail i [7]. The case = 7 is treated i [7]. The case = is kow already to occur i piwheel tiligs. This paper is orgaised as follows. Sectio cotais some basic defiitios ad facts o substitutio tiligs. Readers familiar with this topic may skip this sectio. I order to show that all tiligs have DTO we eed a result o the irratioality of certai agles. This is provided i Theorem 5 i Sectio 3. The costructio of the substitutio rules is give i Sectio 4. Theorem is the a cosequece of Propositios 6, 7, 8, ad 9 i Sectio 4.. Basics For the purpose of this paper a tile is a oempty compact set T R which is the closure of its iterior. A tilig of R is a collectio of tiles T = {T i i N} that is a coverig (i.e. i N T i = R ) as well as a a packig (i.e. the itersectio of the iteriors of ay two distict tiles T i ad T j is empty). A fiite subset of T is called a patch of T. A tilig T has fiite local complexity with respect to rigid motios (FLC for short) if for ay r > 0 there are oly fiitely may pairwise o-cogruet patches i T fittig ito a ball of radius r. (I may other cotexts oe may replace o-cogruet by ot traslates of each other, but here the first optio is the appropriate oe.) A tilig T is operiodic, if T + t = T (t R ) implies t = 0. I additio, T is called aperiodic if each tilig i the hull of T is operiodic. The hull of the tilig T i R is the closure of the set {xt x G} i the local topology. Usually oe takes G = R regarded as traslatios actig o T, or G the group of all rigid motios i R. I our case it does ot matter which oe of the two we choose, see [8]. The local topology ca be defied via a metric. I this metric two tiligs are ε-close if they agree o a large ball of radius ε aroud the origi, possibly after a small motio (e.g. a traslatio by less tha ε). If T arises from a primitive substitutio σ oe may as well speak of the hull of σ, sice all tiligs geerated by σ defie the same hull. See for istace [0, 5, 3,, 8] for more details. A substitutio rule σ is a simple method to geerate operiodic tiligs. A substitutio rule cosists of several prototiles T,..., T m, a iflatio factor λ > ad for each i =,..., m a dissectio of λt i ito cogruet copies of some of the prototiles T,..., T m. The patch resultig from the dissectio is deoted by σ(t i ). A substitutio σ ca be iterated o the resultig patch, by iflatig the patch by λ ad dissectig all tiles accordig to σ. Hece it makes sese to write σ (T i ) or σ k (T i ). A simple example is the substitutio for the piwheel tilig show i Figure. This substitutio uses just oe prototile. The iflatio factor is λ = 5. Oe may as well formulate the piwheel substitutio for two prototiles: if we distiguish a tile ad its mirror image the the piwheel substitutio σ P has two prototiles T ad T (where T is the mirror image of T ), the substitutio σ P (T ) is the mirror image of σ P (T ). I certai istaces we wat to cosider cogruet tiles i T as differet. This is achieved by markigs or colours. Two tiles are equivalet if they are cogruet ad have the same markig or colour. See Subsectio 4.4 below for a example where we cosider cogruet prototiles as differet (e.g. T 3, T 4, T 5 ), with a differet substitutio for each prototile. Give a substitutio σ with prototiles T,..., T m a patch of the form σ(t i ) is called a supertile. More geerally, a patch of the form σ k (T i ) is called a k-th order supertile. A substitutio rule is called primitive if there is k N such that each k-th order supertile cotais cogruet copies of all prototiles.

3 SUBSTITUTION TILINGS WITH DENSE TILE ORIENTATIONS AND FOLD ROTATIONAL SYMMETRY 3 Equivaletly oe may defie primitivity of a substitutio by a associated matrix. The substitutio matrix of a substitutio σ with prototiles T, T,, T m is M σ := (a ij ) i,j m, where a ij is the umber of tiles equivalet to T i i σ(t j ), i, j {,,..., m}. The substitutio is primitive if there is some power of the substitutio matrix cotaiig positive etries oly. Primitivity is a importat property for substitutios. Oe reaso is the followig result, the Perro-Frobeius theorem. Theorem ([4]). Let M be a primitive o-egative square matrix. The M has a real eigevalue λ > 0 which is simple. Moreover, λ > λ for ay eigevalue λ λ. This eigevalue λ is called Perro-Frobeius-eigevalue or PF-eigevalue for short. Furthermore, the associated left ad right eigevectors of λ ca be chose to have positive etries. Such eigevectors are called the left PF-eigevector ad right PF-eigevector of M. Applied to a substitutio tilig T the Perro Frobeius theorem has the followig cosequeces, see for istace [5, 3]. Theorem 3. Let σ be a primitive substitutio i R with iflatio factor λ ad prototiles T, T,, T m ; let M σ be the substitutio matrix of σ. The the PF-eigevalue of M σ is λ. The left PFeigevector cotais the areas of the differet prototiles, up to scalig. The ormalised right PF-eigevector v = (v, v,, v m ) T of M σ cotais the relative frequecies of the prototiles of the tilig i the followig sese: The etry v i is the relative frequecy of T i i T. 3. A irratioality result A agle θ [0, π[ is called irratioal if θ / πq. The piwheel tiligs have ideed DTO because of the fact that the secod order supertile cotais two cogruet tiles which are rotated agaist each other by a irratioal agle (see Figure right, the two tiles are marked by dots). The agle here is arcta(/). It is kow that arccos( ) / πq for 3 odd []. Hece arcta( ) = ( π arccos( 5 )) is irratioal. By iductio the etire tilig cotais tiles that are rotated agaist each other by arcta( ) mod π for all N. Sice arcta( ) is irratioal these values are dese o a circle. More geerally we have the followig result: Theorem 4 ([5, Propositio 3.4]). Let σ be a primitive substitutio i R with prototiles T, T,..., T m. Ay substitutio tilig i the hull of σ has DTO if ad oly if there are k, i such that σ k (T i ) cotais two equivalet tiles T ad T that are rotated agaist each other by some irratioal agle. Hece the desired substitutios eed to ivolve some irratioal agles. The followig result provides such agles. The authors believe that this result must be kow already, but we are ot aware of ay referece. Theorem 5. Let P be a parallelogram with edge legths ad ad iterior agles π 3. The the agles betwee the loger diagoal of P ad the edges of P are irratioal. ( )π ad, h α α π π P P π h h α P h Figure. The agle α (left); α is smaller tha π (middle); α is bigger tha π. Proof. Embed P i the complex plae such that the lower left corer of P coicides with the origi ad the lower base lies alog the real axis as show i Figure. So the upper left corer of P coicides with the poit ξ := e πi/ ad the upper right corer coicides with the poit z = ξ +. Let α be the agle betwee the log diagoal of P ad the x-axis, see Figure left. We will show that α is irratioal.

4 4 BIELEFELD, ATENEO, AND ATENEO Figure 3. Dissectio of regular -gos ito a small regular -go T (), triagles T () ad possibly several parallelograms. Suppose α is ratioal. The there is m such that z m R, which yields ( ) m ( ) m ( ) z z z m ( z ) m = ± = = =, z z zz z hece z z is some m-th root of uity. Because z = ξ + Q(ξ ) we have z z Q(ξ ). It is kow (compare [, Exercise.3]) that roots of uity i Q(ξ ) are of the form ±ξ, k 0 k. Hece z z = ±ξk for some k N. The α = arg ( ) ( ) z z = arg ±ξ k, ad so α = kπ kπ if is eve, ad α = if is odd. For eve cosider a secod parallelogram P with vertices 0,, + ξ, ξ (compare Figure middle). Sice π is the agle betwee ad the diagoal of P we get 0 < α < π, yieldig a cotradictio for eve. For odd cosider a third parallelogram P with vertices 0, + ξ, + ξ + h, h, where h is the legth of the log diagoal of P (compare Figure right). The agle betwee the log diagoal of P ad the real axis is π π. Sice h > we obtai α >. Together with the reasoig above we get π > α > π, yieldig a cotradictio for odd. Therefore α is irratioal. 4. Costructio of the substitutio tiligs The geeral idea for the substitutios is to choose oe prototile as a bisected parallelogram from Theorem 5. To be more precise, for 3 odd the prototile T () is the triagle with iterior agle π where the two edges formig this agle have legth oe ad two, respectively. For 4 eve the prototile T () is the triagle with iterior agle π where the two edges formig this agle have legth oe ad two, respectively. By Theorem 5 the other two agles of this triagle are irratioal for ay 3. A short computatio yields the legth λ of the logest edge as follows: cos( π ) if is odd λ = cos( π ) if is eve. Let λ be the iflatio factor for the desired substitutios σ i the sequel. A regular -go of side legth λ ca be dissected ito copies of T () (alog its edges), oe regular -go with uit edge legth (i its cetre), ad, if 5, ito several parallelograms. This dissectio is illustrated i Figure 3 for the cases 3 9. I order to costruct the desired substitutio tiligs with DTO beig ivariat uder -fold rotatio oe chooses a first prototile T () to be a regular -go of uit edge legth. The substitutio of T () is the give by the dissectio i Figure 3. Therefore the iflatio factor equals λ. If oe ca fid a substitutio for all further prototiles arisig i this dissectio these substitutios are good cadidates for DTO tiligs sice by Theorem 5 the cetral -go of edge legth is rotated agaist the big -go by a irratioal agle. Furthermore give a substitutio exists for some the dissectio of λ T () already provides a tilig ivariat uder -fold rotatio (give a substitutio for all tiles exists at all) sice it may serve as a seed for a fixed poit of σ with T () i the cetre. (To be precise, oe eeds to defie σ icludig a rotatioal part i order to take care of the differet orietatios of the large ad the small regular -gos.)

5 SUBSTITUTION TILINGS WITH DENSE TILE ORIENTATIONS AND FOLD ROTATIONAL SYMMETRY 5 (3) T (3) T T (4) T (4) Figure 4. The substitutios σ 4 (right) ad σ 3 (left). For tiles with o-trivial symmetry the arrows idicate the chirality of the tiles. By choice, all symmetric tiles i the images are right-haded copies. 4.. The 3-fold ad 4-fold tiligs. The substitutios for {3, 4} eed oly two prototiles. Two possible substitutios σ 3 ad σ 4 are show i Figure 4. Propositio 6. For {3, 4} holds: The substitutio σ is a primitive substitutio with DTO. The hull of σ cotais two aperiodic tiligs ivariat uder -fold rotatio. Ay tilig i the hull of σ is FLC with respect to rigid motios. Proof. Obviously the substitutios are primitive, the substitutio matrices are M σ3 = ( ) ad M σ4 = ( ) 3 4. Theorems 5 ad 4 imply that the tiligs have DTO as follows. Let α deote the smallest iterior agle of T (3). Figure 5 shows the situatio for σ 3, the grey shaded tile o the boudary of σ 3 3(T (3) ) is rotated by α agaist the grey shaded tile i the cetre. Sice α is irratioal by Theorem 5, DTO of the tilig i the hull of σ 3 follows by Theorem 4. This pheomeo appears i all substitutios cosidered here ad i the sequel: copies of T () are lied up alog the boudary of σ (T () ). Mirror images of T () are lied up alog the boudary of σ (T () ) (sice they are mirror images Theorem 4 does ot apply here already), ad rotated copies of T () are lied up alog the boudary of σ 3 (T () ). The boudaries of σ (T () ) ad σ 3 (T () ) are rotated agaist each other by α, thus the triagles T () are rotated agaist each other by α. I order to show that the tiligs are aperiodic it suffices to show that the substitutio σ has a uique iverse o the hull [0, Theorem 0..], see also [, 3]. I the cases {3, 4} this is particularly simple: For = 3 ote that each isolated regular triagle T (3) is cotaied i a supertile σ 3 (T (3) ), hece the supertiles σ 3 (T (3) ) ca all be idetified uiquely. The remaiig part of the tilig cosists of supertiles σ 3 (T (3) ), ad the patches of four coected T (3) determie the exact locatio ad orietatio of these supertiles. A similar reasoig works for = 4. Let R α deote the rotatio about the origi through α. Let T () be cetred i the origi. The R α σ (T () ) cotais T () i its iterior. Cosequetly, (R α σ ) k (T () ) cotais (R α σ ) k (T () ) i its iterior. (Figure 5 shows (R α σ 3 ) k (T (3) ) for k = 0,,, 3.) Hece ( (R α σ ) k (T () ) ) k N is a ested sequece that coverges i the local topology. The limit is a tilig T that is fixed uder R α σ. Sice the tiligs i the hull have DTO the hull is ivariat uder rotatios. Thus the patches (R α σ ) k (T () ) are legal i the sese that they are cotaied i tiligs i the hull. Hece T is ideed cotaied i the hull. Sice mirror images of all tiles occur also i all tiligs i the hull, the mirror image of T is also cotaied i the hull, yieldig a secod tilig ivariat uder -fold rotatio. We sketch why the tiligs have FLC with respect to rigid motios. The simplest way to see this is to itroduce a additioal (pseudo-)vertex at the midpoit of the edge of legth i T (). Takig ito accout this pseudo-vertex the tiligs are vertex-to-vertex. A stadard argumet implies that the tiligs are FLC. (There are fiitely may ways how two tiles ca touch each other, hece there are oly fiitely may possible patches fittig ito a ball of radius r. A complete proof of FLC

6 6 BIELEFELD, ATENEO, AND ATENEO Figure 5. Three iteratios of R α σ 3 o T (3). The third order supertile σ3(t 3 (3) ) cotais two copies of T (3) that are rotated agaist each other by a irratioal agle. would eed a list of all possibilities how two tiles ca touch each other, e.g. a list of all vertex stars. Such a list is cotaied i [7] for = 3, 4. More details i [7].) 4.. The 6-fold tilig. For = 6 we get the iflatio factor λ 6 = cos( π 6 ) = 7. The substitutio σ 6 is show i Figure 6. We eed to itroduce a additioal tile T (6) 4 i order to esure primitivity: λ 6 T (6) 3 ca be dissected ito cogruet copies of T (6) ad T (6) 3, but the T (6) would ot occur i ay of the supertiles of T (6) ad T (6) 3, hece the substitutio would ot be primitive. (6) (6) (6) (6) Figure 6. The substitutio σ 6. The substitutio σ 6 has the substitutio matrix M σ6 = , which is primitive because Mσ k 6 cotais oly positive etries for all k 3. The correspodig PF-eigevalue of M σ6 is λ 6 = 7 with left PF-eigevector (6,,, 7) ad ormalised right PFeigevector (, 4, 7, )T. By Theorem 3 the left PF eigevector cotais the areas of the tiles up to scalig, the ormalised right PF-eigevector cotais the relative frequecies of the tiles. The latter ca serve as a startig poit for computig the frequecy module of the tiligs. See [7] for the computatio of the frequecy module of σ 4 by these meas.

7 SUBSTITUTION TILINGS WITH DENSE TILE ORIENTATIONS AND FOLD ROTATIONAL SYMMETRY 7 Propositio 7. The substitutio σ 6 is a primitive substitutio with DTO. The hull cotais two aperiodic tiligs ivariat uder 6-fold rotatio. Ay tilig i the hull of σ 6 is FLC with respect to rigid motios. Proof. The proofs of DTO, FLC ad the existece of a tilig ivariat uder 6-fold rotatio are very much alog the lies i the proof of Propositio 6. Aperiodicity of the tiligs ca be prove similarly by idetifyig the first order supertiles uiquely: the tile T (6) 4 occurs oly as the supertile σ 6 (T (6) 3 ). A patch of six coected hexagos T (6) occurs oly i the supertile σ 6 (T (6) 4 ). All remaiig hexagos T (6) determie the supertiles σ(t (6) ). Everythig that remais are supertiles σ 6 (T (6) ). For more thorough proofs see [7] The 8-fold tilig. For = 8 we get the iflatio factor λ 8 = cos( π 8 ) = 5 +. The substitutio rule is show i Figure 7. Sice λ 8 T (8) 3 ad λ 8 T (8) 4 caot be dissected ito copies of the prototiles T (8), T (8), T (8) 3, T (8) 4 we eed to itroduce itermediate tiles T (8) 5 := λ 8 T (8) 3 ad T (8) 6 := λ 8 T (8) 4 i order to defie a substitutio rule. Note that we eed to defie a orietatio o the tiles i order to distiguish a tile from its mirror image. This is ot idicated i the figure. There are several possibilities to do so. Oe possibility is lettig all tiles i figure have the same orietatio. The oly poit where this really matters is that the tile T (8) i σ 8 (T (8) ) has the same orietatio as the prototile T (8) i order to esure DTO. T (8) T 3 (8) T 4 (8) T 5 (8) T 6 (8) T (8) Figure 7. The substitutio σ 8. Orietatios ad chiralities of symmetric tiles are arbitrary if ot show i the image. Propositio 8. The substitutio σ 8 is a primitive substitutio with DTO. The hull cotais two aperiodic tiligs ivariat uder 8-fold rotatio. Proof. Agai the proofs of DTO ad the existece of a tilig ivariat uder 8-fold rotatio are very much alog the lies i the proof of Propositio 6. Aperiodicity of the tiligs ca be prove similarly as above by idetifyig the first order supertiles uiquely. More details i [7]. The primitivity ca be checked via the substitutio matrix M σ8 = M 4 σ 8 has oly positive etries. We suppose that the tiligs i the hull of σ 8 have also FLC. But sice the tiligs with or without pseudo-vertices are ot vertex-to-vertex (this ca be see i σ8(t 3 (8) ) for istace) a rigorous proof will be rather legthy. For details we refer to [7].

8 8 BIELEFELD, ATENEO, AND ATENEO 4.4. The 5-fold tilig. It is possible to defie the desired substitutio for = 5 usig just six prototiles. However the tiligs may ot have FLC. I order to esure FLC we defie a substitutio σ 5 with prototiles. The iflatio factor is λ 5 = The substitutio rule is show i Figure 8. T T T 6 T 7 T T T 3 T 8 T 4 T 9 T 5 T 0 Figure 8. The substitutio σ 5. Half arrows idicate the orietatio of edges ad tiles. All coicidet edges have the same orietatio. Propositio 9. The substitutio σ 5 is a primitive substitutio with DTO. The hull cotais two aperiodic tiligs ivariat uder 5-fold rotatio. Ay tilig i the hull of σ 5 is FLC with respect to rigid motios. Proof. As before the proofs of DTO ad the existece of a tilig ivariat uder 5-fold rotatio are alog the lies of the proof of Propositio 6. FLC follows from the fact that coicidet edges have the same orietatio ad that edges of the same legth are dissected i the same maer uder σ 5. Aperiodicity of the tiligs ca be prove similarly by idetifyig the first order supertiles uiquely. More details i [7]. Primitivity of σ 5 ca be checked by cosiderig the substitutio matrix M σ5 below. Sice Mσ 5 5 cotais oly positive etries the substitutio σ 5 is primitive.

9 SUBSTITUTION TILINGS WITH DENSE TILE ORIENTATIONS AND FOLD ROTATIONAL SYMMETRY 9 The substitutio matrix for σ 5 looks as follows M σ5 = The powers of M σ5, the PF-eigevalue ad the PF-eigevectors have bee computed with the computer algebra software (CAS) Scietific Workplace 5.5 [8]. The PF-eigevalue is (equals λ 5, as it ought to be) ad its correspodig left PF-eigevector ad ormalised right PF-eigevector are respectively give by T w = ad v = Agai the left PF eigevector w cotais the areas of the tiles up to scalig, the right PFeigevector v cotais the relative frequecies of the tiles. 5. Coclusio We were ot able to defie a geeral substitutio rule σ for all, or at least for all eve. Ayway, there is a geeral patter for dissectig λ T () ( 4) ito five copies of T () ad four further triagles, ad for dissectig λ T () ito oe copy of T (), copies of T () ad several parallelograms. Hece it is likely that there are substitutio tiligs with DTO ivariat uder -fold rotatio for all 3. This might be also of iterest with respect to a commet i []:...there is a lack of kow examples of aperiodic plaar tilig families with higher orders of rotatioal symmetry.. That paper cotais the first substitutio tilig with elevefold symmetry appearig i the literature. Our method might yield further primitive substitutio tiligs with -fold ad also -fold rotatioal symmetry (though the umber of prototiles might be huge). The proof of Theorem 5 o irratioal agles i cyclotomic parallelograms uses the fact that the cosidered irratioal agle α is smaller tha π. Hece the result geeralises immediately to the log diagoals of parallelograms with iterior agle π ad with edge legths a b Q rather tha ad. Usig other argumets it might be possible to show the irratioality of other agles as well, e.g. the agle of the short diagoal. For the sake of briefess we did ot metio further implicatios of our costructios i the cotext of dyamical properties of the hull. Just to metio a few: the fact that a tilig has FLC esures the miimality of the hull of σ. Due to primitivity of σ all tiligs i the hull of σ are repetitive, hece have uiform patch frequecies. As a cosequece we obtai: If we deote the

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