Aperiodic Tilings: from the Domino Problem to an Aperiodic Monotile

Aperiodic tilings: from the Domino problem to an aperiodic monotile

Mike Whittaker (University of Glasgow)

Western Sydney University Centre for Research in Mathematics and Data Science Abend

4 June 2020 Plan

1. The Domino Problem

2. Searching for small aperiodic tile sets

3. Socolar and Taylor’s monotile

4. A dendritic monotile

5. An orientational monotile 1. The Domino Problem

H. Wang, Proving theorems by pattern recognition, II, Bell Sys. Tech. J. 40 (1961), 1–41. R. Berger, The Undecidability of the Domino Problem, Memoirs AMS 66, Providence, 1966. B. Gr¨unbaumand G.C. Shephard, Tilings and Patterns, W.H. Freeman, New York, 1987. E. Jeandel and M. Rao, An aperiodic set of 11 Wang tiles, preprint (2015), arXiv1506.06492. S. Labb´e, Substitutive structure of Jeandel-Rao aperiodic tilings, preprint (2018), arXiv1808.07768. Tilings

2 A prototile is a subset of R that is homeomorphic to a closed disc, along with labels.

2 Let P be a ﬁnite set of prototiles. A tiling of R is a countable collection of tiles T = {ti : i ∈ N} such that: 2 ti = p − x for some p ∈ P and x ∈ R ; [ 2 ti = R ; i∈N int (ti ) ∩ int (tj ) = ∅ if i 6= j. A patch is a connected ﬁnite collection of tiles from a tiling T . A tiling is non-periodic if T + x = T implies x = 0. SUBSTITUTIVE STRUCTURE OF JEANDEL-RAO APERIODIC TILINGS

SÉBASTIEN LABBÉ

1. Introduction Aperiodic tilings are much studied for their beauty but also for their links with various aspects of science includings dynamical systems [Sol97], topology [Sad08], theoretical computer science [Jea17]The and Domino crystallography. Problem Chapters 10 and 11 of [GS87] and the more recent book on aperiodic order [BG13] give an excellent overview of what is known on aperiodic tilings. The first examples of aperiodicHao tilings Wang were proposed tilings oftheZ2 Dominoby Wang Problem tiles, that in is, 1961: unit square tiles with colored edges [Ber65,Knu69,Rob71,Kar96,Cul96,Oll08]. A tiling by Wang tiles is valid if every contiguous edges have the sameA color. finite The set set of of prototiles all valid tilings are called using aWang finiteset tiles (orof Wang Wang tiles is called the Wang shift of and denoted . It is a 2-dimensional subshift as it isT invariant under translations and dominoes)T if they are isometric squares, and their edges are closed underT taking limits. A nonempty Wang shift is said to be aperiodic if none of its tilings marked by specific colours or symbols.T have a nontrivial period.

4 2 1 2 1 1 0 1 2 2 3 2 2 2 2 3 1 3 1 3 3 3 0 0 0 0 3 1 0 1 1 1 3 1 0 1 2 3 1 1 2 2 4 2

Figure 1. The Jeandel-Rao’s set of 11 Wang tiles. T0

arXiv:1808.07768v1 [math.DS] 23 Aug 2018 Tiles in a Wang tiling can meet along an edge only if their Jeandel andsymbols Rao [JR15] match. proved that the set of Wang tiles shown in Figure 1 is the smallest possible set of Wang tiles that is aperiodic. Indeed, based on computer explorations, they proved that every WangThe tile Domino set of cardinality Problem asks10 either whether admits there a periodic is an algorithm tiling of the to plane or does not tile the plane at all. Thus there is no aperiodicÆ Wang tile set of cardinality less than or equal to 10. In the same work,decide they whether found this a set interesting of Wang candidate tiles can of tile cardinality the plane. 11 and they proved that it is aperiodic. TheirWang proof also is showed based on that the descriptionit is possible of a to sequence ﬁnd a set of transducers of Wang describing larger and larger inﬁnite horizontal strips by iteratively taking product of themselves. Their example is also minimal fortiles the that number mimics of colors. the behaviour Indeed it is of known any Turingthat three machine. colors are not enough to allow an aperiodic tile set [CHLL12] and Jeandel and Rao mentionned in their preprint that while they

Date: August 24, 2018. 2010 Mathematics Subject Classiﬁcation. Primary 52C23; Secondary 37B50. Key words and phrases. Wang tiles and tilings and aperiodic and substitutions and markers. 1 The Domino Problem

If a set of Wang tiles can tile the plane, then one of these possibilities must hold: the set admits only periodic tilings, for example just one regular (undecorated) square. the set admits periodic and non-periodic tilings, for example two squares labelled a and b. the set admits only non-periodic tilings, such a set of prototiles is called aperiodic.

Wang showed that the Domino Problem is decidable if and only if there does not exist an aperiodic set of Wang tiles, and conjectured that no such set exists.

The ﬁrst aperiodic set of Wang tiles was found by Wang’s student Robert Berger in 1966, and consisted of 20,426 tiles. This was ﬁrst reduced by Berger to 104 tiles, then by Knuth to 96 tiles. SUBSTITUTIVE STRUCTURE OF JEANDEL-RAO APERIODIC TILINGS

SÉBASTIEN LABBÉ

Abstract. We describe the substitutive structure of Jeandel-Rao aperiodic Wang tilings 0. We introduce twelve sets of Wang tiles i 1 i 12 together with their associated Wang shifts {T } Æ Æ i 1 i 12. Using a method proposed in earlier work, we prove the existence of recognizable 2- { } Æ Æ dimensional morphisms Ê : for every i 0, 1, 2, 3, 6, 7, 8, 9, 10, 11 that are onto up to i i+1 æ i œ{ } a shift. Each Êi maps a tile on a tile or on a domino of two tiles. We also prove the existence of a topological conjugacy ÷ : which shears Wang tilings by the action of the matrix ( 11) and 6 æ 5 01 an embedding fi : that is unfortunately not onto. The Wang shift is self-similar, ape- 5 æ 4 12 riodic and minimal. Thus we give the substitutive structure of a minimal aperiodic Wang subshift X0 of the Jeandel-Rao tilings 0.ThesubshiftX0 ( 0 is proper due to some horizontal fracture Theof 0’s Domino or 1’s in tilings Problem in and we believe that X is a null set. Algorithms are provided 0 0 \ 0 to find markers, recognizable substitutions and sheering topological conjugacy from a set of Wang tiles. From Grunbaum and Shephard’s 1987 book Tilings and Patterns (p.596): “The reduction in the number1. Introduction of Wang tiles in an aperiodic Aperiodic tilingsset from are much over studied20,000 for to their16 has beauty been but a notable also for theiracheivment. links with various aspects of science includingsPerhaps dynamical the minimum systems possible [Sol97], number topology has [Sad08], now been theoretical reached. computer science [Jea17] and crystallography.If, however, further Chapters reductions 10 and 11 are of [GS87] possible and then the more it seems recent book on aperiodic order [BG13] give an excellent overview of what is known on aperiodic tilings. The first examples of aperiodic tilingscertain were that tilings new of ideasZ2 by andWang methods tiles, that will is, be unit required.” square tiles with colored edges [Ber65,Knu69,Rob71,Kar96,Cul96,Oll08]. A tiling by Wang tiles is valid if every contiguous edges have the sameThe color. smallest The set setof all of valid Wang tilings tiles using is now a finite 11, set providedof Wang by Jeandeltiles is called the Wang shift of andand denoted Rao in. 2015. It is a They2-dimensional also showedsubshift thatas no it isT such invariant set exists under for translations and T T closed under takingn ≤ 10. limits. A nonempty Wang shift is said to be aperiodic if none of its tilings have a nontrivial period. T

4 2 1 2 1 1 0 1 2 2 3 2 2 2 2 3 1 3 1 3 3 3 0 0 0 0 3 1 0 1 1 1 3 1 0 1 2 3 1 1 2 2 4 2

Figure 1. The Jeandel-Rao’s set of 11 Wang tiles. T0 arXiv:1808.07768v1 [math.DS] 23 Aug 2018 Jeandel and Rao [JR15] proved that the set of Wang tiles shown in Figure 1 is the smallest possible set of Wang tiles that is aperiodic. Indeed, based on computer explorations, they proved that every Wang tile set of cardinality 10 either admits a periodic tiling of the plane or does not tile the plane at all. Thus there is no aperiodicÆ Wang tile set of cardinality less than or equal to 10. In the same work, they found this interesting candidate of cardinality 11 and they proved that it is aperiodic. Their proof is based on the description of a sequence of transducers describing larger and larger inﬁnite horizontal strips by iteratively taking product of themselves. Their example is also minimal for the number of colors. Indeed it is known that three colors are not enough to allow an aperiodic tile set [CHLL12] and Jeandel and Rao mentionned in their preprint that while they

SÉBASTIEN LABBÉ

1. Introduction Aperiodic tilings are much studied for their beauty but also for their links with various aspects of science includings dynamical systems [Sol97], topology [Sad08], theoretical computer science [Jea17] and crystallography. Chapters 10 and 11 of [GS87] and the more recent book on aperiodic order [BG13] give an excellent overview of what is known on aperiodic tilings. The ﬁrst examples of aperiodic tilings were tilings of Z2 by Wang tiles, that is, unit square tiles with colored edges [Ber65,Knu69,Rob71,Kar96,Cul96,Oll08]. A tiling by Wang tiles is valid if every contiguous edges have the same color. The set of all valid tilings using a ﬁnite set of Wang tiles is called the Wang shift of and denoted . It is a 2-dimensional subshift as it isT invariant under translations and T closed underT taking limits. A nonempty Wang shift is said to be aperiodic if none of its tilings The Domino Problem T have a nontrivial period.

4 2 1 2 1 1 0 1 2 2 3 2 2 2 2 3 1 3 1 3 3 3 0 0 0 0 3 1 0 1 1 1 3 1 0 1 2 3 1 1 2 2 4 2

SUBSTITUTIVE STRUCTURE OF JEANDEL-RAO APERIODIC TILINGS 3 Figure 1. The Jeandel-Rao’s set 0 of 11 Wang tiles. 2 2 2 1 2 2 2 2 3 2 2 2 T2 2 2 2 1 2 2 2 1 1 1 1 1 0 0 3 3 1 1 1 1 1 1 1 1 3 3 1 1 1 1 1 1 1 1 1 1 1 1 0 0 3 3 1 1 1 1 1 arXiv:1808.07768v1 [math.DS] 23 Aug 2018 4 4 2 2 2 4 4 4 2 2 4 4 4 4 4 2 2 2 4 4 4 4 2 2 2 4 4 4 2 2 4 4 4 4 4 2 2 2 4 4 Jeandel and Rao2 2 2 [JR15]2 2 2 2 2 proved2 2 2 2 that2 2 2 the2 2 2 set2 2 of2 2 Wang2 2 2 2 tiles2 2 2 shown2 2 2 2 in2 2 Figure2 2 2 2 12 is the smallest 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 possible set of Wang1 1 tiles0 that0 0 is aperiodic.1 1 1 0 Indeed,0 1 based1 1 on1 computer1 0 0 explorations,0 1 1 they proved 0 3 3 0 0 0 0 0 0 0 0 3 3 3 3 0 0 0 0 0 0 3 3 3 3 0 0 3 3 0 0 0 0 0 0 0 0 3 3 3 that every Wang2 tile1 set1 of1 cardinality1 2 3 110 1either1 2 admits3 1 a periodic2 1 1 tiling1 1 of the2 plane3 or does not 2 1 1 1 1 2 3 1 1 1 2 3 1 2 1 1 1 1 2 3 1 0 0 3 3 0 0 3 3 3 3 1 1 3Æ3 0 0 3 3 3 3 1 1 3 3 1 1 0 0 3 3 0 0 3 3 3 3 1 1 3 tile the plane at all.2 2 Thus1 there2 3 is2 no2 aperiodic1 2 Wang3 2 tile2 set1 of2 cardinality2 1 2 3 less2 than2 or equal to 10. 2 2 1 2 3 2 2 1 2 3 2 2 1 2 2 1 2 3 2 2 In the same work,3 1 they1 0 0 found3 3 1 1 this3 3 1 interesting1 0 0 3 3 1 1 candidate3 3 1 1 0 0 of3 3 cardinality1 1 0 0 3 3 111 1 and3 3 1 they1 0 proved that it 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 is aperiodic. Their2 2 2 proof2 2 2 is2 based2 2 2 2 on2 2 the2 2 description2 2 2 2 2 2 2 of2 2 a2 sequence2 2 2 2 2 2 of2 transducers2 2 2 2 2 2 2 describing2 larger 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 and larger inﬁnite0 horizontal0 0 0 strips0 0 by0 iteratively0 0 0 taking0 0 0 product0 0 of0 themselves.0 0 0 0 Their example is 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 also minimal for the1 1 number1 1 of1 colors.1 1 Indeed1 1 it1 is known1 1 1 that1 three1 1 colors1 1 are1 not1 enough to allow 3 3 3 0 0 3 3 0 0 3 3 3 3 0 0 3 3 0 0 3 3 3 3 0 0 3 3 3 3 0 0 3 3 0 0 3 3 3 3 0 an aperiodic tile3 set1 [CHLL12]2 1 2 and3 Jeandel1 2 1 and2 Rao3 mentionned1 2 3 1 in2 their1 preprint2 3 1 that while they 3 1 2 1 2 3 1 2 1 2 3 1 2 3 1 2 1 2 3 1 1 3 3 1 1 0 0 3 3 1 1 3 3 1 1 0 0 3 3 1 1 3 3 3 3 1 1 3 3 1 1 0 0 3 3 1 1 3 3 3 2 1 2 2 2 2 1 2 2 2 2 3 2 2 1 2 2 2 2 3 Date: August 24,2 2018.1 2 2 2 2 1 2 2 2 2 3 2 2 1 2 2 2 2 3 1 0 0 3 3 1 1 1 1 1 1 0 0 3 3 1 1 1 1 1 1 1 1 3 3 1 1 0 0 3 3 1 1 1 1 1 1 1 1 3 2010 Mathematics2 Subject2 2 Classiﬁcation.4 4 2 2 Primary2 4 52C23;4 4 Secondary2 2 2 37B50.2 2 4 4 4 2 2 2 2 4 4 2 2 2 4 4 4 2 2 2 2 2 4 4 4 2 Key words and phrases.2 2 2 2 2 Wang2 2 2 tiles2 2 2 and2 2 tilings2 2 2 2 and2 2 aperiodic2 2 2 2 2 and2 2 substitutions2 2 2 2 2 2 2 and2 2 markers.2 2 2 2 2 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 0 0 0 1 1 1 2 1 1 1 1 2 3 1 1 1 1 1 1 2 3 1 1

Figure 3. A finite part of a Jeandel-Rao aperiodic tiling in 0. Any tiling in the minimal subshift X0 of 0 that we describe can be decomposed uniquely into 19 supertiles (two of size 45, six of size 72, four of size 70 and seven of size 112). The thick black lines show the contour of the supertiles. The figure illustrates two complete supertiles of size 72 and 45. While Jeandel-Rao tile set is not self-similar, the 19 supertiles corresponding to are T0 T12 self-similar. The figure illustrates the presence of a horizontal fracture of 0’s which implies that not all tilings in 0 are the image of a tilings in 12 by a sequence of substitutions. However, we believe that almost all of them are.

We show that the map fi : is not onto. The problem comes from the existence of tilings 5 æ 4 in 0, 1, 2, 3 and 4 that have horizontal fractures of 0’s or 1’s. The tilings in 4 obtained by sliding along the fracture line can not be obtained as the image under the application of fi of a tiling in 5. But we believe that fi is onto up to a set of measure 0. Thus if this conjecture holds, we give a complete characterization of the substitutive structure of tilings 0 made with Jeandel-Rao’s tiles. The article is structured as follows. In Section 2, we present the necessary definitions and notations on Wang tiles including self-similarity, recognizability and aperiodicity. In Section 3, we recall the definition of markers, the desubstitution of Wang shifts and we propose algorithms to compute them. In Section 4, we desubstitute tilings from 0 to 4. In Section 5, we show we can remove two tiles from Õ to get Õ. In Section 6, we construct the tile set by adding T4 T4 µT4 T5 decorations to tiles 4 to avoid fracture lines in tilings of 4 and we prove that fi :5 4 is an embedding. In SectionT 7, we construct the shearing topological conjugacy ÷ : æ .In 6 æ 5 Section 8, we desubstitute tilings from 6 to 12. In Section 9, we prove the main results. In Section 10 we contruct eight tilings in 12 that are fixed by the square of Ê12. In Section 11, we show that 4 X4 is nonempty. In Section 12, we list in a table the tile sets i, markers Mi i and morphisms\ for every i with 0 i 12. T µT i+1 æ i Æ Æ 2. The search for small aperiodic tile sets

R. Penrose, Pentaplexity: A class of non-periodic tilings of the plane, Math. Intelligencer 2 (1979), 32–37. Relaxing the rules At this point, the search for other aperiodic sets of prototiles began. Speciﬁcally, we allow tiles to be any shape, and we no longer count rotations. In the 1970s Roger Penrose found a set of aperiodic prototiles with just two prototiles!

Interestingly, combining these tiles in various ways gives rise to a set of Wang tiles with only 24 tiles (which would have been the record at the time had anyone noticed). See Exercise 11.1.2 in Gr¨unbaumand Shephard. A patch of the Penrose tiling Kleenex Toilet Paper

Penrose tiling embossed toilet paper “So often we read of very large companies riding rough-shod over small businesses or individuals, but when it comes to the population of Great Britain being invited by a multi-national to wipe their bottoms on what appears to be the work of a Knight of the Realm without his permission, then a last stand must be made.” - David Bradley, director of Pentaplex (the company that cares for Penrose’s copyrights) The einstein (one stone) problem

Penrose’s aperiodic two prototile set leads to the obvious question: is an aperiodic monotile possible?

But what are the rules for a single prototile? The ultimate einstein would

not allow reﬂections of the monotile, have a monotile that is homeomorphic to a closed disc, and have matching rules that are forced by shape alone.

Whether such a monotile exists is still an open problem. Computer searches have been running for at least a decade, with no success yet. However, it sounds as though Rao is turning his attention to this problem... so who knows. Further relaxations of the rules

One can ask for an einstein with less stringent rules. Here are some that the experts have deemed reasonable. An einstein could

allow reﬂections of the monotile, not be homeomorphic to a closed disc, have matching rules that are not forced by shape alone; for example, colour matching rules with more than one colour allowed to meet, or have matching rules that reach beyond adjacent tiles. (a) (b) (c)

(a) (b) (c)

Figure 1. The hexagonal prototile and its mirror image with color matching rules. (a) The two tiles are related by reflection about a vertical line. (b) Adjacent tiles must form continuous black stripes. Flag decorations at opposite ends of a tile edge, such as the indicated flags at opposite ends of the vertical edge, must point in the same direction. (c) A portion of an infinite Figure 1. The hexagonal prototile and its mirror image with color matching rules. (a) The two tiling that respects the matching rules. tiles are related by reflection about a vertical line. (b) Adjacent tiles must form continuous black stripes. Flag decorations at opposite ends of a tile edge, such as the indicated flags at opposite ends of the vertical edge, must point in the same direction. (c) A portion of an infinite tiling that respects the matching rules.

Figure 2. The SCD prototile and the space–filling tiling it forces. tiles must form continuous black stripes, and flag decorations FigureThe 2. uniformlyThe SCD prototile space-filling, and the three-dimensionalspace–filling tiling it forces. prototile at opposite ends of each tile edge must point in the same of Figure 2, a rhombic biprism, was exhibited by Schmitt, direction. (The arrows in (b) point to the two flagstiles at must opposite form continuousConway, black and stripes, Danzer and flag [10 decorations]. To fill space,The one uniformly is forced space-filling, to three-dimensional prototile ends of a vertical tile edge.) Each tile in (c) is a rotationat opposite and/or ends ofconstruct each tile edge 2D periodic must point layers in the of same tiles sharingof Figure triangular2, a rhombic faces, biprism, was exhibited by Schmitt, reflection of the single prototile, and the only waydirection. to fill space (The arrowswith in (b) ridges point running to the two in flags the at direction opposite of oneConway, pair of and rhombus Danzer [10]. To fill space, one is forced to ends of a vertical tile edge.) Each tile in (c) is a rotation and/or construct 2D periodic layers of tiles sharing triangular faces, while obeying the rules everywhere is to form a nonperiodic, edges on top and the other pair below. The layers are then reflection of the single prototile, and the only way to fill space with ridges running in the direction of one pair of rhombus hierarchical extension of the pattern in (c). while obeying the rulesstacked everywhere such that is to each form is a nonperiodic, rotated by an angleedges/ onwith top respect and the to other pair below. The layers are then hierarchical extensionthe of one the pattern below in it, (c). where / is the acute anglestacked of such the that rhombic each is rotated by an angle / with respect to Defining the einstein base. Any choice of / other than integerthe multiples one below of it,p/3 where or / is the acute angle of the rhombic Two constructions that could conceivably beDefining counted the as einsteinp/4 produces a tiling that is not periodic,base. and Any certain choice choices of / other than integer multiples of p/3 or einsteins were discovered in 1995. A singleTwo prototile constructions that thatpermit could a tiling conceivably in which be counted the number as ofp/4 nearest produces neighbor a tiling that is not periodic, and certain choices forces a pattern of the Penrose type was presentedeinsteins were by discoveredenvironments in 1995. is A finite, single so prototile that the that prototilepermit can a be tiling endowed in which the number of nearest neighbor Gummelt (with a complementary proof by Steinhardtforces a pattern and ofwith the Penrosebumps and type nicks was inpresented a way that by locksenvironments the relative is finite, posi- so that the prototile can be endowed with bumps and nicks in a way that locks the relative posi- Jeong) [8, 9]. But in this case tiles are allowed toGummelt overlap (with and a complementarytions of adjacent proof layers. by Steinhardt and tions of adjacent layers. Jeong) [8, 9]. But in thisAgain, case tiles however, are allowed the to universal overlap and reaction was ‘‘This is not the covering of the space is not uniform. For thisthe reason covering the of the space is not uniform. For this reason the Again, however, the universal reaction was ‘‘This is not prototile is not considered to be an einstein. prototile is not consideredreally to what be an we einstein. are looking for.’’ The nonperiodicityreally what we areof the looking for.’’ The nonperiodicity of the ...... 3. The Socolar–Taylor monotile

JOSHUA E. S. SOCOLAR received his JOSHUA E. S. SOCOLARJOAN M.received TAYLOR his took up mathematicsJOAN in M. TAYLOR took up mathematics in Ph.D. in Physics from the University of Ph.D. in Physics from1991 the at University age 34 after of being inspired1991 by at a age 34 after being inspired by a Pennsylvania in 1987, with a thesis on magazine article on quasicrystals featuring

Pennsylvania in 1987, with aAUTHORS thesis on magazine article on quasicrystals featuring

AUTHORS quasilattices and quasicrystals. He has been Penrose’s rhombus tiling. She began but did quasilattices and quasicrystals. He has been Penrose’s rhombus tiling. She began but did on the faculty at Duke since 1992, and he is not complete a degree, preferring to conduct on the faculty at Duke since 1992, and he is afﬁliated with the Centernot complete for Nonlinear a degree, and preferring to conducther own research. Since then she has pursued afﬁliated with the Center for Nonlinear and Complex Systems andher the own Duke research. Center for Since then she has pursuedtiling and related topics in abstract algebra and Complex Systems and the DukeJ.E.S Socolar Center and J.M. for Taylor, An aperiodic hexagonal tile,tiling J. and related topics in abstract algebra and Comb. Th. A 118 (2011), 2207–2231.Systems Biology. His hobbies include jazz number theory including original work on Systems Biology. His hobbiesJ.E.S include Socolar and J.M. jazz Taylor,piano,Forcing singing, nonperiodicity and with wordnumber a single games. theory including original workconstructible on polygons. She likes to unwind piano, singing, and word games.tile, Math Intel. 34 (2012), 18–28. constructible polygons. She likes to unwindwith knitting and reading. J.M. Taylor, Aperiodicity of a functional monotile, preprint (2010); available from Physics Departmentwith and Centerknitting for and reading. Physics Department and Centerhttp://www.math.uni-bielefeld.de/sfb701/preprints/view/420. for Nonlinear and Complex Systems Post Office Box U91 Nonlinear and Complex Systems Duke University Post Office Box U91 Burnie, TAS 7320 Durham, NC 27708 Australia Duke University USA Burnie, TAS 7320 Durham, NC 27708 e-mail: [email protected] USA e-mail: [email protected] Ó 2011 Springer Science+Business Media, LLC, Volume 34, Number 1, 2012 19

Ó 2011 Springer Science+Business Media, LLC, Volume 34, Number 1, 2012 19 The Socolar–Taylor monotile

The existence of an aperiodic monotile was resolved almost a decade ago by Joshua Socolar and Joan Taylor. 2208 J.E.S. Socolar, J.M. Taylor / Journal of Combinatorial Theory, Series A 118 (2011) 2207–2231

Fig. 1. The prototile and color matching rules. (a) The two tiles shown are related by reflection about a vertical line. (b) Adjacent tiles must form continuous black stripes. Flag decorations at opposite ends of a tile edge (as indicated by the arrows) must point in the same direction.The (c) following A portion of rules an infinite must tiling. hold (For ain color any image, given the readertiling: is referred to the web version of this article.) R1 : the black curves must be continuous across all edges in the tiling, and R2 : the flags at the vertices of two tiles separated by a single tile edge must always point in the same direction.

Fig. 2. Alternative coloring of the 2D tiles.

1. The prototile

Aversionoftheprototile,withitsmirrorimage,isshowninFig.1.Therearetwoconstraints,or “matching rules,” governing the relation between adjacent tiles and next-nearest neighbor tiles:

(R1) the black stripes must be continuous across all edges in the tiling; and (R2) the ﬂags at the vertices of two tiles separated by a single tile edge must always point in the same direction.

The rules are illustrated in Fig. 1(b) and a portion of a tiling satisfying the rules is shown in Fig. 1(c). We note that this tiling is similar in many respects to the 1 ϵ ϵ2 tiling exhibited previously by + + Penrose [1]. There are fundamental differences, however, which will be discussed in Section 6. For ease of exposition, it is useful to introduce the coloring scheme of Fig. 2 to encode the matching rules. The mirror symmetry of the tiles is not immediately apparent here; we have replaced left-handed flags with blue stripes and right-handed with red. The matching rules are illustrated on the right: R1 requires continuous black stripes across shared edges; and R2 requires the red or blue segments at opposite endpoints of any given edge and collinear with that edge to be different colors. The white and gray tile colors are guides to the eye, highlighting the different reflections of the prototile. Throughout this paper, the red and blue colors are assumed to be merely symbolic indicators of the chirality of their associated flags. When we say a tiling is symmetric under reflection, for example, we mean that the flag orientations would be invariant, so that the interchange of red and blue is an integral part of the reflection operation. The matching rules presented in Figs. 1 and 2 would appear to be unenforceable by tile shape alone (i.e., without references to the colored decorations). One of the rules specifying how colors must match necessarily refers to tiles that are not in contact in the tiling and the other rule cannot be implemented using only the shape of a single tile and its mirror image. Both of these obstacles can be overcome, however, if one relaxes the restriction that the tile must be a topological disk. Fig. 3 shows how the color-matching rules can be encoded in the shape of a single tile that consists of several disconnected regions. In the figure, all regions of the same color are considered to compose asingletile.Theblackstriperuleisenforcedbythesmallrectanglesalongthetileedges.Thered– blue rule is enforced by the pairs of larger rectangles located radially outward from each vertex. The flag orientations (or red and blue stripe colors) are encoded in the chirality of these pairs. (For a discussion of the use of a disconnected tile for forcing a periodic structure with a large unit cell, see [2].) The Socolar–Taylor monotile The matching rules on the Taylor–Socolar tile: require reflections of the monotile and Figure 5. The partial translational symmetryhave matching with the rules smallest that spacing. reach Clustersbeyond of adjacent 24 shaded tiles. tiles (two of each of the twelveHowever, tile orientations) if one are is repeated not concerned throughout whether the tiling, the forming monotile a is a triangular lattice. Purple stripestopological are shown only disc, for then a subset the of matching one third of rules the tiles. can be forced by shape alone: (a) (b)

Figure 6. (a) Enforcing by the shape alone with a disconnected 2D tile. All the patches of a single color, taken together, form a single tile. (b) A deformation of the disconnected prototile in (a) to a prototile with cutpoints.

The hexagonal blocks on each arm have thickness h/3, consists of two square projections that fill opposite quadrants allowing the blocks from three crossing arms to make a full of the hole; the other type fills the entire hole but only to half column. The six arms on the prototile have outer faces that its depth. The two types are both invariant under rotation by are tilted from the vertical in a pattern that encodes the chi- 180°. Two plugs of the same type can fit together to fill a hole, rality of the flags of the 2D tile. Forming one triangular lattice but plugs of different types cannot. Finally, we place two requires that bevels of opposite type be joined, and hence columns of three plugs each on each of the large vertical faces that flags of opposite chirality match in accordance with R2. of the main hexagonal portion of the tile. Each column The small bumps on the tiles and the holes in the arms are aligned with a black stripe has plugs of one type, and the arranged such that adjacent tiles can fit together if and only if other columns have plugs of the other type. (The latter are the black stripes match up properly, as required by R1. The needed to fill the holes in the arms at those positions.) Three three square holes in each arm are positioned so that pro- plugs are needed because of the staggered heights of jections from the faces on neighboring tiles can meet with neighboring tiles. If a prototile that is a topological sphere is each other. The holes are all the same; they do not themselves desired, the plugs can be moved toward the middle of their encode the positions of the black stripes. Next, we create two respective faces so that the left and right side plugs meet and types of plug that can be inserted into a hole. One type the holes in the arms are converted to U-shaped slots.

Ó 2011 Springer Science+Business Media, LLC, Volume 34, Number 1, 2012 23 Socolar–Taylor monotile Theorem (Socolar–Taylor 2010) The Socolar–Taylor monotile is aperiodic; that is, there are tilings satisfying R1 and R2, and every such tiling is nonperiodic. Sketch of proof: Recall the local rule: R1 : the black curves are continuous across all edges in the tiling. Notice that R1-lines always combine to form nested triangles:

0 1 3 7 Socolar–Taylor monotile

Start with a tiling by hexagons. We will add R1-curves until all tiles have R1-curves that satisfy R1

Look at all possible three tile clusters of tiles satisfying R1.

2210 WeJ.E.S. see Socolar, that everyJ.M. Taylor possible / Journal cluster of Combinatorial has twosmall Theory, corners Series A 118 (2011) 2207–2231 appearing together.

The only possible tile that can be placed beside two corners is a third corner, and hence any tiling satisfying R1 must contain a 0-triangle.

Fig. 4. The forced pattern of small black rings. (a, b) There must be at least one black ring. (c) A black ring at one vertex of a tile forces a black ring at the other vertex. (d) The forced periodic partial decoration. (For clearer visualization of the red and blue colors, the reader is referred to the web version of this article.) we show that at least one tiling does indeed exist by giving a constructive procedure for ﬁlling the plane with no violation of the matching rules.

Proof. It is immediately clear that the hexagonal tiles (without colored markings) can fill space to form a triangular lattice of tiles. We begin with such a lattice of unmarked tiles and consider the possibilities for adding marks consistent with R1 and R2.Notefirstthattheconfigurationofdark black stripes in Fig. 4(a) requires the completion of the small black ring indicated by the gray stripe. Second, as illustrated in Fig. 4(b), attaching a long black stripe to a portion of a ring immediately forces the placement of a decoration (gray) that leads to the formation of a small ring. Thus the tiling must contain at least one small black ring. Inspection of the matching rules for tiles adjacent to a single tile reveals that if a small black ring is formed at one vertex of the original tile, there must be a small black ring at the opposite vertex. Fig. 4(c) shows the reasoning. Note that the positions of the “curved” black stripes on the prototile determine the orientations of the red and blue diameters (but not the red–blue diameter). Given the central tile with a small black ring at its lower vertex, the vertical diameter of the tile at the lower left must be red. The tile at the upper left is then forced to be (partially) decorated as shown. R2 requires that the vertical stripe be blue. It cannot be just the bottom half of a red–blue diameter because that would not allow the black stripes to match. The vertical blue diameter forces the creation of another small black ring at the top of the original tile. Applying the same reasoning to all the tiles in Fig. 4(c) and iteratively applying it to all additional forced tiles, shows that the honeycomb lattice of small black rings and colored diameters shown in Fig. 4(d) is forced. The locations of the longer black stripes and the orientation of the red–blue diameter of each tile are not yet determined. We refer to the tiles that are partially decorated in this figure as level 1 (L1) tiles. If a tile is placed in one of the open positions in Fig. 4(d), the local structure shown in Fig. 5 must be formed. Consider now the relation between two tiles that fill adjacent holes in Fig. 4(d). The dashed lines in Fig. 5 show the edges of larger hexagons that can be thought of as larger tiles. Consider first the black stripes on a large tile. Because of the forced black stripes on the L1 neighbors of the added tile, the black stripes on the large tile have the same form as those on the L1 tiles and thus the large tiles obey R1. (Note that adding the long black stripe on the central tile will force corresponding long black stripes on the L1 tiles to its left and right.) Similarly, the forced orientations of the red–blue diameters on the L1 next-nearest neighbors act to transfer R2 to the large tiles. For the large tiles, the matching of opposite colors is forced because each large tile edge is a red–blue Socolar–Taylor monotile

Now rule R2 is designed to ensure that if the corners of two R1-triangles meet at a common tile, then they must have the same length. Since we have ensured that there must be at least one length 0 triangle, R2 therefore implies that there must be a hexagonal grid of length 0 triangles: Socolar–Taylor monotile

An analogous argument shows that at least one length 1 triangle exists. So R2 implies that we obtain a hexagonal grid of length 1 triangles. Socolar–Taylor monotile

Continuing this process ad inﬁnitum gives a tiling of the plane that satisﬁes R1. Socolar–Taylor monotile

Each of these hexagonal grids of length 2n − 1 triangles is a lattice with periodicity constant 2n+1.

However, there is no largest size, so the resulting tiling must 2208 J.E.S. Socolar, J.M. Taylor / Journal of Combinatorial Theory,not Series have any A translational 118 (2011) periodicity; 2207–2231 that is, the tiling is nonperiodic.

Fig. 2. Alternative coloring of the 2D tiles.

1. The prototile

Aversionoftheprototile,withitsmirrorimage,isshowninFig.1.Therearetwoconstraints,or “matching rules,” governing the relation between adjacent tiles and next-nearest neighbor tiles:

(R1) the black stripes must be continuous across all edges in the tiling; and (R2) the ﬂags at the vertices of two tiles separated by a single tile edge must always point in the same direction.

M. Mampusti and M. Whittaker, A monotile that forces nonperiodicity through a local dendritic growth rule, preprint, to appear in Bull. LMS. A dendritic monotile Goal: Deﬁne a monotile that does not require reﬂections and each tile only interacts with adjacent tiles.

Tilings must be constructed as follows. Once a single tile has been placed, a direct isometry of the monotile can be added to the plane provided the resulting collection of tiles is a patch, and R1 the oﬀ-centre black lines and curves must be continuous across tiles (´ala Socolar–Taylor) and R2 the new tile’s red tree continuously connects with at least one red tree of an adjacent tile. A dendritic monotile

Note that R1 is the same rule as used for the Socolar–Taylor monotile, however, since we do not require a reﬂected copy it can be realised by shape:

We say that a tiling T satisﬁes growth rule R if every patch in T is contained in a patch that can be constructed following rule R.

Therefore, a tiling T satisﬁes R2 if and only if the union of R2-trees in T is connected. A dendritic monotile

The following two patches are legal: A dendritic monotile

Theorem (Mampusti-W, 2019) Our dendritic monotile is aperiodic; that is, there are tilings satisfying R1 and R2, and every such tiling is nonperiodic.

I will sketch the proof of existence, which follows from constructing an inﬁnite nested union of patches. A dendritic monotile

P0 P1 P2

P3 Constructing a tiling

Fix P0 at the origin, then we obtain a tiling of the plane: ∞ [ T := Pn n=1 5. An orientational monotile

J. Walton and M. Whittaker: An aperiodic tile with edge-to-edge orientational matching rules, preprint. An orientational monotile Goal: Find a monotile with similar properties to the Socolar-Taylor monotile, but with edge-to-edge matching rules.

Two tiles t1 and t2 are permitted to meet along a shared edge e only if: R1 the off-centre black lines and curves must be continuous across e (ála Socolar–Taylor) and R2 whenever the two charges at e in t1 and t2 both have a clockwise orientation then they must be opposite in charge. The drawback is that we allow more flexibility with what counts as a ‘rule’ for edge matchings. An orientational monotile

Theorem (Walton–W, 2019) Our orientational monotile is aperiodic; that is, there are tilings satisfying R1 and R2, and every such tiling is nonperiodic.

In this case, I want to sketch the proof that any tiling constructed from this tile is nonperiodic.

(time permitting) Proof of nonperiodicity

Recall that R1 lines form inﬁnite lines or triangles.

Deﬁne anticlockwise to be the standard orientation of R1 triangles, and then assign R1 segments the same charge as the unique clockwise charge ‘inside’ its triangle.

Notice that straight R1 lines are assigned charges consistently. We let ch(E) ∈ {+, −} denote the charge of an R1 edge. Proof of nonperiodicity

Edges enforce charges of those they ‘lead’ to (notation: E1 a E2):

N F E1 a E2 ⇒ ch(E1) = ch(E2), E1 a E2 ⇒ ch(E1) = (−1) · ch(E2) F F F This immediately implies that no chain E1 a E2 a E3 a E1 of 3 edges. Proof of nonperiodicity So edges that meet ‘far’ must lead to longer edges, or else we could create a spiral leading to a cycle of 3 edges. Proof of nonperiodicity So edges that meet ‘far’ must lead to longer edges, or else we could create a spiral leading to a cycle of 3 edges. Proof of nonperiodicity So either: a) there must be R1 triangles of arbitrarily large size or b) there must be an inﬁnite R1 line. We get nonperiodicity immediately in case a) Case b) For an inﬁnite R1 line we always have the following pattern of triangles above and below the line.