Tiling groups for Wang tiles

Transcription

1 Tiling goups fo Wang tiles Cistophe Mooe Ivan apapot Eic émila Abstact We apply tiling goups and height functions to tilings of egions in the plane by Wang tiles, which ae squaes with coloed boundaies whee the colos of shaed edges must match. We define a set of tiles as unambiguous if it contains all tiles equivalent to the identity in its tiling goup. Fo all but one set of unambiguous tiles with two colos, we give efficient algoithms that tell whethe a given egion with coloed bounday is tileable, show how to sample andom tilings, and how to calculate the numbe of local moves o flips equied to tansfom one tiling into anothe. We also analyze the lattice stuctue of the set of tilings, and study seveal examples with thee and fou colos as well. Intoduction Tilings of the plane with Wang tiles [, 8] have been studied in compute science since the famous esult of ege [4] that the poblem of whethe we can tile the infinite plane using a given set of Wang tiles is undecidable. This pape focuses on tilings of a given finite egion with coloed bounday. This is a well-known NP-complete poblem [5, ] and we intend to tackle the subpoblem in which the numbe of colos is fixed. Ou appoach is algebaic: This wok was patially suppoted by pogams FONDECYT 9966 (I..), FONDAP on Applied Mathematics and ECOS, and the Sandia Univesity eseach Pogam (C.M.) Univesity of New Mexico and the Santa Fe Institute, NM, USA mooe@cs.unm.edu DIM and CMM, Univesidad de Chile, Chile iapapot@dim.uchile.cl LIP, École Nomale Supéieue de Lyon, and IMA, IUT oanne, Fance eemila@ens-lyon.f we use the tiling goups of Conway and Lagaias [7], and height functions, intoduced by Thuston [26] and independently in the statistical physics liteatue (see [6] fo a eview). These ideas wee used and genealized by Kenyon and Kenyon [3], émila [23, 25], Popp [2], and othes, fo the poblem of tiling plana egions with diffeent types of polyominoes o simple polygons. Ou wok is, to ou knowledge, the fist time Wang tiles have been addessed with these techniques. We define a set of tiles as unambiguous if a cetain algebaic condition is both necessay and sufficient fo single tiles. Fo all but one unambiguous set of two-colo tiles, we give a polynomial-time algoithm to tell whethe a given egion with given colos on its bounday is tileable. We also study the stuctue of the set of tilings unde local flips that change the colo of a few inteio edges, and show that this is eithe a distibutive lattice o a hypecube. In paticula, this gaph is connected, i.e. any tiling can be tuned into any othe with a seies of flips, and we give a fomula fo the minimum numbe of flips necessay to do so. In seveal cases, these tilings tun out to be equivalent to familia systems with height functions, such as domino tilings and Euleian oientations. We can then apply the techniques of Luby, andall and Sinclai [6] and Popp and Wilson [22] to sample andom tilings in polynomial time. We finish by caefully studying a set of tiles with thee colos, and by noting that some sets of thee- and fou-colo tiles possess two- and theedimensional height functions. We also note that seveal sets of tiles ae isomophic in the sense that thee is a natual bijection between pais of tilings and bounday conditions, even though thei tilings goups ae not isomophic.

2 2 The tiling goup Let Λ be the squae lattice of the Euclidean plane 2. A (finite) egion P of Λ is a (finite) union of closed squae cells of Λ. A egion P is said to be a polygon if its inteio and its complement 2 \ P ae both connected. A Wang tile is a squae of side one with coloed edges. An assignment of Wang tiles to the cells of a polygon P coesponds to a tiling if tiles on neighboing cells have the same colo along thei common edge. Thoughout the pape, the bounday conditions of a tiling will include not just the shape of the egion, but the colos on its bounday. Let S be a set of Wang tiles constucted with two colos, lue and ed. Let P be a polygon with colos and on the edges of its bounday. We study the poblem of finding a tiling of P using the tiles of S in such a way that the colos of the bounday condition ae satisfied. Let W = w, w 2, w 3, w 4, w 5, w 6 } be the set of all Wang tiles with two colos. i.e. w w 2 w 3 w 4 w 5 w 6 Note that we allow these tiles to be otated. A subset w x,, w xn } W will be denoted by W x x n. To solve the tiling poblem fo paticula subsets of W, we stat by intoducing an oientation of the edges of Λ. Fist, we will assume that the squaes of Λ ae coloed black and white like a checkeboad. We oient the edges of Λ so that they go clockwise and counteclockwise aound black and white squaes espectively, so that an ant going along an edge will have a white squae on its left and a black squae on its ight. Wheneve we have a tiling T of a polygon P with tiles in W, the colos of the edges of P will be eithe o. Let us wite a symbol b wheneve we move along a blue edge with the oientation, and b when we move against it. Similaly, we wite and fo moving along a ed edge. To evey tiled polygon P with coloed edges we can associate a contou wod w b,, b, } stating fom any extenal vetex and following a path aound the bounday of P. Let S W and let v be the set of contou wods of the tiles in S. Then the tiling goup S = b, v is the fee goup modulo the elations w = e fo each contou wod w v. This can also be witten as a facto S = b, /N S whee N S is the nomal subgoup geneated by the contou wods in v and thei conjugates. Note that uv = e if and only if vu = e, and that fo squae two-colo tiles, evey mio image is also a otation. Thus it doesn t matte whee we stat on a tile, o in which diection we go aound it, to define its contou wod; we obtain the same tiling goup S. On the othe hand, fo thee o moe colos, we would have to explicitly allow eflections as well as otations. Now that we have defined the tiling goup fo tiles with coloed edges, we make seveal obsevations. Fist, any tiling T of a polygon P with a set of tiles S coesponds to a tiling function f T : V S, whee V is the set of vetices in P o on its bounday. We do this by fist fixing f on a paticula vetex, say f T (x ) = e, whee x is the leftmost vetex of the bottom of P. We then define f inductively as follows: If we have aleady assigned an element x S to a vetex v and if the oiented edge (v, u) P is coloed with (esp. ), then set f T (u) = xb (esp. f T (u) = x). Similaly, if (u, v) P is coloed (esp. ) then set f T (u) = xb (esp. x ). Thus moving along the aows, o against them, changes the value of f T (v) by b,, b, o. If b in S then the map fom tilings to tiling functions is invetible, since we can get the colo of any edge in T by compaing f T at its ends. It is easy to pove by induction on the numbe of cells that f T (v) is well-defined; it is single- 2

3 valued since going aound any single tile gives a contou wod which is equivalent to the identity of S. The same obsevation gives Poposition (Conway s citeion) If a polygon P with a coloed bounday admits a tiling with a set of tiles S, then its contou wod is equivalent to the identity in S. The convese of this poposition is obviously false fo some sets of tiles, even fo egions consisting of a single cell! This leads us to the notion of an unambiguous set of tiles. A set S is unambiguous if the convese of Conway s citeion is tue fo single cells, i.e. a tile belongs to S if and only if its contou wod is e S. Fo instance, if S is unambiguous then it cannot contain two tiles which diffe only in the colo of a single edge unless S = W, since dividing the contou wod of one of these tiles by the othe gives = b. These and simila consideations easily show that the unambiguous subsets of W ae the singletons, W 2, W 3, W 23, W 6, W 25, W 34, W 56, W 23, W 24, W 234, and W. These ae the sets of tiles we will focus on next. Poposition 2 Conway s citeion is sufficient fo a set of tiles S if and only if (i) S is unambiguous and (ii) Fo any thee colos x, y, z, thee is at least one colo w such that the tile with edges x, y, z, w (going, say, counteclockwise) is in S. Poof. To pove the fist diection, making Conway s citeion sufficient fo single squaes is the definition of unambiguity. Since the contou wod of the domino shown in the figue below is xx y z zy = e, sufficiency implies that it must be tileable, meaning that thee must exist a colo w fo the inteio edge. To pove the convese, note that if the second condition holds, we can take any polygon, stat at the bounday, and epeatedly choose cells which can be emoved while keeping the egion simply-connected as in Muchnik and Pak [9]. Since each such cell has at most thee of its edges set by the bounday conditions, condition (ii) allows us to place a tile thee and emove it fom the egion, until only one cell emains. This cell is tileable if and only if it is in S, which if S is unambiguous means if and only if Conway s citeion holds. Y Z X It is easy to see that this ules out all sets except W 56, W 234, and W, fo which the eade can easily check both conditions. We discuss these futhe below. 3 Unambiguous two-colo tiles 3. Tivial cases Fo W, W 2, W 3, W 4, W 2, W 3, W 23 and W 23, a polygon has at most one tiling, and if it exists we can find it in time popotional to the aea. This is because fo all these sets the tile is detemined by the colos of two adjacent edges, so we can stat at a cone of P and scan, say, left to ight and top to bottom. Tivially any polygon can be tiled if S = W, and Conway s citeion is tivially sufficient with W = 4. We note as well that the numbe of such tilings is 2 m whee m is the numbe of edges in the inteio of P. If we define a local flip as changing the colo of an edge (and the two tiles on eithe side of it), then the set of tilings has the stuctue of an m-dimensional hypecube. We will see below that simila stuctues can be found fo othe unambiguous sets of tiles. 3.2 Finite tiling goups: W 56 and W 234 Poposition 2 shows that Conway s citeion is sufficient as well as necessay fo W 56 and W 234. This gives us a linea-time algoithm fo tileability: simply calculate P s contou wod and compae it to the identity. The tiling goup of W 56 is b, b 3, 3 b. Since b 3 = e, we have = b 3, and since b = 3 = b 9, the goup is isomophic to 8 with (say) b = and = 3. Fo W 234, on the othe Z X Y 3

4 hand, since bb = bb = e we have b = b, and since b 4 = 4 = b 2 2 = e, the tiling goup is isomophic to 4 2 with (say) = (, ) and b = (, ). Although thei tiling goups ae not isomophic, thee is a simple bijection between tilings with the two sets. y flipping the colos of the hoizontal edges on the even-numbeed ows (say), we change one edge of each tile, tansfoming tiles in W 56 to tiles in W 234 and vice vesa. This may also change some of the colos on the bounday, so this is actually a bijection between pais (P, T ) whee P is a egion with coloed bounday and T is a tiling of it. Thee is also a simple bijection between these and assignments of two colos, say yellow and geen, to the vetices of P. If we colo an edge ed if its two vetices ae the same colo and blue if they ae diffeent, we obtain tilings with the set W 234. This also shows that, once the colos on P s bounday ae chosen, the numbe of tilings is 2 k whee k is the numbe of vetices in the inteio of P. The local flip changes the colo of a vetex and thus the fou edges and the fou tiles aound it, and the gaph of tilings foms a k-dimensional hypecube. This gives a tivial algoithm fo sampling andom tilings, by flipping k independent coins. 3.3 Infinite tiling goups and height functions The use of height functions fo tilings was intoduced by Thuston [26], and since then have been applied to seveal sets of polyominoes and polygons. They had been found independently in statistical physics, and have been applied to ice models, antifeomagnets, and Potts models (see e.g. [3, 6]). We will see that they can be applied to some sets of Wang tiles as well. The idea is to tansfom the tiling function f T to an intege height at each vetex, by composing it with an appopiate function z : S and witing h T = z f T. Then we can define a patial ode on the set of tilings of a paticula polygon with coloed bounday, T T v P : h T (v) h T (v) The height function typically possesses the following popeties, which will help us solve tiling poblems: iven the bounday conditions, thee is a one-to-one elation between tilings and height functions. Local flips can be applied at local extema of h T in the inteio of P, and can connect any tiling to any othe with the same bounday conditions. With espect to, the the set of tilings is a distibutive lattice. In paticula, thee ae minimal and maximal tilings and. is convex, i.e. h has no local maxima in the inteio of P. The distibutive lattice stuctue will help us in seveal ways. Since thee exists a tiling iff thee exists a minimal tiling and the heights of the vetices of the bounday ae given by the bounday conditions, this will give us a staightfowad algoithm fo tileability. We will also have an algoithm to compute the shotest way to pass fom a tiling to anothe one by flips. Finally, the technique of coupling fom the past will allow us to sample andom tilings in polynomial time [6, 22] W 5 and dominoes It is easy to see that if S is the singleton W 5 we have domino tilings, whee blue edges ae the boundaies of dominoes and ed edges coss thei inteios. The tiling goup b, b 3 is isomophic to with b = and = 3. Thus we can take the height function h T = f T whee z is simply the identity. If the peimete of P is blue, Conway s citeion simply checks that thee ae an equal numbe of black and white squaes in P. We aleady know fom Poposition 2 that this citeion is not sufficient. To discuss the lattice stuctue of the set of tilings we will follow [24] and omit the poofs. A local flip can be applied at a vetex v when its two incoming edges have the same colo, 4

5 and its two outgoing edges have the same colo. Equivalently, a flip consists of exchanging two hoizontal dominoes fo two vetical ones o vice vesa. It is easy to see that a flip is possible at v if and only if v is a local extemum of the height function. Since this flip changes the colo of all fou edges aound a vetex, it can only be applied at a vetex in the inteio of P. We call a flip upwads if it tansfoms a local minimum to a local maximum and downwads if it does the evese. The eade can check that these flips incease o decease h T (v) by 4. ecall that a lattice [5, 9] is a set equipped with a patial ode, whee any two elements a and b have a unique infimum a b and a unique supemum a b. A lattice is distibutive if a (b c) = (a b) (a c) and a (b c) = (a b) (a c). Viewing the set as a diected acyclic gaph, it follows that any pai of diected paths between two points (which, if a path exists, ae compaable) have the same length. Standad aguments then allow us to pove the following popeties of the patial ode defined above: Poposition 3 (Flips and ode) Fo any pai of tilings T and T of the same polygon P with the same coloed bounday, T T if and only if T can be obtained fom T by a sequence of upwads flips. Poposition 4 (Lattice stuctue) If a polygon P is tileable then, with espect to the ode, the gaph of tilings is a distibutive lattice. In paticula, thee is a unique minimal tiling. Poposition 5 (Convexity) Let P be a tileable polygon and let be its minimal tiling. Then h has no local maximam in the inteio of P. Poposition 6 (Flip fomula) Fo any pai of tilings T and T with the same bounday conditions, the minimal numbe of flips to pass fom T to T is (/4) v h T (v) h T (v). Combining Popositions 4 and 5 gives the following algoithm, which eithe constucts the minimal tiling o confims that the egion is not coloable: Calculate the heights of vetices on the bounday. If Conway s citeion is not satisfied then the egion is not tileable. Othewise epeat the following steps until the egion is completely tiled. Wheneve all fou cones of a cell have an assigned height, place the appopiate tile thee and emove that cell fom the egion. If no tile is consistent with these heights, halt and conclude that the egion is not tileable. Find a vetex v with maximum height h max on the cuent bounday; it has a neighbo w whose height has not yet been assigned. Since h may not have local maxima in the inteio, set h (w) smalle than h max, eithe to h max o h max 3 depending on the oientation of the edge fom v to w. If we emove cells at P s bounday wheneve one of thei edges is ed, tilings with W 5 coespond exactly to domino tilings of the emaining egion. We can then use the esults of Popp and Wilson [22] and Luby, andall and Sinclai [6] to sample pefectly andom tilings in time polynomial in the aea of P W 34, W 24, and Euleian oientations The tiling goup of W 34 is b, bb, b 2 2, which is isomophic to with b = + and =. Once again we can take h T = f T as ou height function. We have all the same tools as in the pevious example, except that now fo any pai of neighbos u, v we have h T (u) h T (v) =. This is ecognizable as the height function fo Euleian oientations of the dual lattice, called the six-vetex ice model by physicists [3]. This is also equivalent to the height function fo thee-coloings of the squae lattice, and to altenating-sign matices [2]. The algoithm fo tileability is completely analogous to that fo the domino tiling W 5, except that we always set 5

6 the height of the neighboing vetex to h max. The pogess of the algoithm is shown in Figue ; it eithe constucts the minimum tiling, o aives at a contadiction whee two neighboing vetices have heights diffeing by moe than, violating the definition of the height function and poving that the egion is not tileable. Fo W 24, the tiling goup b, b 4, 4, b 2 2 is moe complex. Howeve, by imposing the additional elations b 2 = 2 = e we can obtain a simple quotient fo it, the fee goup on two geneatos of ode 2, which has the following Cayley gaph: 2 b b b e b b b b While this is not isomophic to it clealy has the same shape as. We can define a height function h T = z f T with the following z, taking advantage of the fact that if 2 = b 2 = e evey element can be witten as a wod w of altenating s and b s: z(w) = w if w begins with b z(w) = w if w begins with Then the same esults follow as fo W 34. Just as fo W 56 and W 234, thee is a simple bijection between tilings with W 34 and those with W 24 even though thei tiling goups ae not isomophic. If we flip the colos of all the hoizontal edges (say), each tile in W 34 becomes one in W 24 and vice vesa. Composing this with the bijections shown above gives a simple bijection between W 24 tilings and Euleian oientations. As in the pevious case, the techniques of [6, 22] can be used to sample andom tilings in polynomial time The cuious case of W 6 The set W 6 (and its symmety patne W 25 ) is the only unambiguous two-colo set which emains unsolved. Its tiling goup b, b 4, 3 b is isomophic to 2 with b = 3 and =. Thus Figue : The pogess of the tiling algoithm fo W 34, o equivalently Euleian oientations of the gid, which constucts the minimal tiling o shows that none exists. 6

7 its tiling goup is finite; howeve, Conway s citeion is not sufficient. y lifting fom 2 to we see that the numbe of w tiles on white squaes minus the numbe of w tiles on black squaes is an invaiant, since fo a polygon P this is n/2 whee n is the intege coesponding to P s contou wod. Notice that, fo each vetex, the tiling function has thee possible values, since f T (v) is equivalent mod 4 to the length of a path fom the oigin vetex to v. We leave as an open poblem whethe thee is a polynomial-time algoithm to tell whethe a given polygon can be tiled with W 6. The eade may enjoy showing that a egion with ed bounday is tileable if and only if it can be tiled by dominoes and X-pentominoes. It seems likely that such tilings ae NP-complete fo non-simply-connected egions using constuctions simila to Mooe and obson [8]. 4 Examples with moe colos 4. Height functions on Cayley tees In this section we show that the notion of height function can also be used fo some Wang tiles with thee (o moe) colos. While the height function is moe complex, it still gives us an efficient algoithm to detemine whethe a egion is tileable. Ou example is a genealization of W 24, whee each tile has at most two colos, and whee evey coloed edge shaes a vetex with anothe edge with the same colo. The set V is the following: Taking ou colos to be ed, een and lue, with the associated geneatos, g and b, then the tiling goup is, g, b 2 g 2, g 2 b 2, b 2 2 (note that these elations imply 4 = g 4 = b 4 = e) and this appeas to be quite complex. Luckily, thee is a simple quotient which does not ceate any ambiguity, namely =, g, b 2, g 2, b 2. Its Cayley gaph Γ() has a tee stuctue (if opposite aows ae identified) as shown in Figue 2, and a height function can be constucted fom the following axioms: Fo evey n, z((b) n ) = 2n and z((b) n b) = 2n. Fo each element x of, thee exists a unique neighbo p (x) of x, called the - pedecesso of x, such that z(p (x)) = z(x). Fo the othe two neighbos y of x, we have z(y) = z(x) +. These imply, fo instance, that if w is a wod in, g, b} whee no two adjacent lettes ae the same, then z(w) = w if w begins with o g z(w) = w 4n if w begins with (b) n g z(w) = w 4n 2 if w begins with (b) n bg To define a patial ode on, we say that x y if thee exists a finite sequence of elements of, stating with x and finishing with y, such that the pedecesso in the sequence is the -pedecesso as defined above. If we define the index of any element x as h(x) mod 2, then the patial ode induces an ode i on each of the two index classes. Each element v has a unique pedecesso in its index class, p i (x) = p (p (x)). Fo each pai x, y of elements with the same index i, we define inf i (x, y) as the infimum of x and y with espect to i. Notice that inf i (x, y) is not always equal to the infimum inf (x, y) with espect to, since the latte might be in the othe index class, in which case inf i (x, y) = p (inf (x, y)). Note also that h T (v) is equivalent mod 2 to the length of any path fom the oigin vetex to v, so f T (v) and f T (v) ae in the same index class fo any two tilings T, T of the same egion. A local flip changes the colos aound a fixed inteio vetex v. This can only happen if all the edges linking v to its neighbos have the same 7

8 b g can be flipped downwads at v, inducing a tiling T such that f T (v) = p i (f T (v)) whee i is the index of f T (v). We have T T T, and d(t, T ) = d(t, T ) 2. This gives the esult by induction, until d(t, T ) = and T = T Figue 2: The Cayley tee Γ() fo the theecolo tiling, and the height function z. colo, which means that the height function has a local extemum at v. As befoe, we define a patial ode on tilings by T T if f T (v) f T (v) fo all v (which implies f T i f T and h T h T ). We also define a distance between elements of : fo x, y, let d(x, y) be the length of the shotest path between them, using the geneatos, g and b. Then define the distance between two tilings as d(t, T ) = v d(f T (v), f T (v)). Note that if T and T ae compaable, we have d(t, T ) = v h T (v) h T. Poposition 7 (Flips and ode) Fo any pai of tilings T and T with the same bounday conditions, T T if and only if T can be obtained fom T by a sequence of downwads flips. Poof. y induction on d(t, T ). Take a vetex v such that h T (v) > h T (v), whee v is a local maximum of h T (note v is an inteio vetex). Then h T (u) = h T (v) and f T (u) = p (f T (v)) fo all neighbos u of v. Thus T 2 3 ecall that an infeio semi-lattice is simila to a lattice, but with only the infimum of two elements a b guaanteed to be unique. Then: Poposition 8 (Infeio semi-lattice stuctue) If a polygon P is tileable then the gaph of tilings with espect to is a infeio semi-lattice, whee the tiling function of T = T T is given by f T = inf i (f T, f T ) at each vetex. Poof. We have to pove that f T = inf i (f T, f T ) is a valid tiling function. Note that if u, v ae neighbos (by which we mean that the edge connecting them is in P ) then thei values of the tiling function ae neighbos in Γ(), i.e. eithe f T (u) = p (f T (v)) o f T (v) = p (f T (u)), and similaly fo T. Theefoe, we need to show that f T (u) and f T (v) ae neighbos in Γ(). We have two cases up to symmety. If f T (u) = p (f T (v)) and f T (u) = p (f T (v)), then if f T (v) and f T (v) ae compaable, then f T (u) = p (f T (v). If f T (v) and f T (v) ae incompaable, then inf (f T (u), f T (u)) = inf (f T (v), f T (v)), in which case eithe f T (u) = p (f T (v)) o f T (v) = p (f T (u)) using the elationship between inf and inf i stated above. The othe case, in which f T (u) = p (f T (v)) and f T (v) = p (f T (u)), can be analyzed similaly. Thus the pointwise infimum of f T and f T with espect to i is a tiling function, and the coesponding tiling T is clealy the unique infimum of T and T with espect to. This also implies that thee is a unique minimal tiling, which has the same popeties as in the simple cases above: Poposition 9 (Convexity) Let P be a egion tileable with tiles in V and let be its minimal tiling. Then h has no local maxima in the inteio of P. 8

9 Poof. Suppose that fo we have an intenal vetex v such that h (v) is a local maximum. Then flipping v downwads would give a new tiling T, a contadiction. This gives us an efficient algoithm fo constucting the minimal tiling and confiming tileability simila to that of Section 3.3., except that we set f (w) = p (f max ). We also have: Poposition (Flip fomula) Fo any pai of tilings T and T satisfying the same bounday condition, the minimal numbe of flips to go fom T to T is (/2)d(T, T ). Poof. We use the method of [25]. Clealy, (/2)d(T, T ) is a lowe bound fo the numbe of necessay flips since flipping at v changes d(t, T ) by zeo o ±2. Now take an inteio vetex v such that f T (v) f T (v) and sup(h T (v), h T (v)) is locally maximal. We assume w.l.o.g. that sup(h T (v), h T (v)) = h T (v), in which case f T (u) = p (f T (v)) fo each neighbo u of v. Then T can be flipped at v, moving f T (v) towads f T (v) in Γ() and giving a tiling T such that d(t, T ) = d(t, T ) 2 (notice that this flip eithe educes the height of v o keeps it the same, changing f T (v) but not h T (v)). This gives the esult by induction. Note that we have actually defined one height function in an uncountably infinite family of them, whee the height deceases along one path (in this case (b) ) and inceases along all othes. Each of these induces a diffeent patial ode, and fo each computable one we have an algoithm simila to that above to find the minimal tiling with espect to it. 4.2 Highe-dimensional height functions As anothe example, conside the set V of foucolo Wang tiles whee a tile is in V if and only if each colo appeas once on its bounday. The tiling goup 4 /,,, } is Abelian and infinite, and is isomophic to the bodycenteed cubic lattice with the fou geneatos (+, +, +), (+,, ), (, +, ), and (,, +). This coesponds to a theedimensional height function. Similaly, theecolo tiangula tiles have a two-dimensional height function 3 /,, } isomophic to the tiangula lattice, and six-colo hexagonal tiles have a five-dimensional height function. All these tilings ae equivalent to edge k- coloings of the dual lattice (the squae, hexagonal, and tiangula lattices espectively) whee k is equal to the dual lattice s degee. Edge 3- coloings of the hexagonal lattice ae also equivalent to vetex 4-coloings of the tiangula lattice, and wee studied by axte [2], Huse and utenbeg [2], and Mooe and Newman [7]. Edge 4-coloings of the squae lattice wee studied by Kondev and Henley [4]. None of these tilings ae connected unde local moves, but they ae connected unde loop moves whee we choose two colos, find a loop consisting of edges with those two colos, and switch the colos along the loop. Little is known about the mixing time of the esulting Makov chain; the techniques of [6, 22] appea not to apply, since these nonlocal moves make it had to define a monotonic coupling. We suggest this as an aea fo futue eseach. efeences [] C. Allauzen and. Duand, Tiling poblems. In E. oge, E. adel, Y. uevich, The classical decision poblem. Spinge-Velag, 997. [2].J. axte, Coloings of a hexagonal lattice. J. Math. Phys. (97) [3] H. van eijeen, Exactly solvable model fo the oughening tansition of a cystal suface. Phys. ev. Lett. 38 (977) [4]. ege, The undecidability of the domino poblem. Memois of the Ameican Mathematical Society 66 (996) [5]. ikhoff, Lattice theoy. Ameican Mathematical Society, 967. [6] J.K. uton J. and C.L. Henley, A constained Potts antifeomagnet model with an inteface 9

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