Aperiodic tilings: from the Domino problem to an aperiodic ...

Aperiodic tilings: from the Domino problem toan aperiodic monotile

Mike Whittaker(University of Glasgow)

Western Sydney UniversityCentre 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, MemoirsAMS 66, Providence, 1966.

B. Grunbaum and 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. Labbe, Substitutive structure of Jeandel-Rao aperiodic tilings,preprint (2018), arXiv1808.07768.

Tilings

A prototile is a subset of R2 that is homeomorphic to a closed disc,along with labels.

Let P be a finite set of prototiles. A tiling of R2 is a countablecollection of tiles T = {ti : i ∈ N} such that:

ti = p − x for some p ∈ P and x ∈ R2;⋃i∈N

ti = R2;

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.

The Domino Problem

Hao Wang proposed the Domino Problem in 1961:

A finite set of prototiles are called Wang tiles (or Wangdominoes) if they are isometric squares, and their edges aremarked by specific colours or symbols.

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 {Ti}1ÆiÆ12 together with their associated Wang shifts{�i}1ÆiÆ12. Using a method proposed in earlier work, we prove the existence of recognizable 2-dimensional morphisms Êi : �i+1 æ �i for every i œ {0, 1, 2, 3, 6, 7, 8, 9, 10, 11} that are onto up toa shift. Each Êi maps a tile on a tile or on a domino of two tiles. We also prove the existence of atopological conjugacy ÷ : �6 æ �5 which shears Wang tilings by the action of the matrix ( 1 1

0 1 ) andan embedding fi : �5 æ �4 that is unfortunately not onto. The Wang shift �12 is self-similar, ape-riodic and minimal. Thus we give the substitutive structure of a minimal aperiodic Wang subshiftX0 of the Jeandel-Rao tilings �0. The subshift X0 ( �0 is proper due to some horizontal fractureof 0’s or 1’s in tilings in �0 and we believe that �0 \ X0 is a null set. Algorithms are providedto find markers, recognizable substitutions and sheering topological conjugacy from a set of Wangtiles.

1. Introduction

Aperiodic tilings are much studied for their beauty but also for their links with various aspectsof 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 aperiodicorder [BG13] give an excellent overview of what is known on aperiodic tilings. The first examplesof 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 edgeshave the same color. The set of all valid tilings using a finite set T of Wang tiles is called the Wangshift of T and denoted �T . It is a 2-dimensional subshift as it is invariant under translations andclosed under taking limits. A nonempty Wang shift �T is said to be aperiodic if none of its tilingshave a nontrivial period.

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

Jeandel and Rao [JR15] proved that the set of Wang tiles shown in Figure 1 is the smallestpossible set of Wang tiles that is aperiodic. Indeed, based on computer explorations, they provedthat every Wang tile set of cardinality Æ 10 either admits a periodic tiling of the plane or does nottile the plane at all. Thus there is no aperiodic 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 itis aperiodic. Their proof is based on the description of a sequence of transducers describing largerand larger infinite horizontal strips by iteratively taking product of themselves. Their example isalso minimal for the number of colors. Indeed it is known that three colors are not enough to allowan 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.

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Tiles in a Wang tiling can meet along an edge only if theirsymbols match.

The Domino Problem asks whether there is an algorithm todecide whether a set of Wang tiles can tile the plane.

Wang also showed that it is possible to find a set of Wangtiles that mimics the behaviour of any Turing machine.

The Domino Problem

If a set of Wang tiles can tile the plane, then one of thesepossibilities must hold:

the set admits only periodic tilings, for example just oneregular (undecorated) square.the set admits periodic and non-periodic tilings, for exampletwo squares labelled a and b.the set admits only non-periodic tilings, such a set of prototilesis called aperiodic.

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

The first aperiodic set of Wang tiles was found by Wang’sstudent Robert Berger in 1966, and consisted of 20,426 tiles.This was first reduced by Berger to 104 tiles, then by Knuthto 96 tiles.

The Domino Problem

From Grunbaum and Shephard’s 1987 book Tilings andPatterns (p.596):

“The reduction in the number of Wang tiles in an aperiodicset from over 20,000 to 16 has been a notable acheivment.Perhaps the minimum possible number has now been reached.If, however, further reductions are possible then it seemscertain that new ideas and methods will be required.”

The smallest set of Wang tiles is now 11, provided by Jeandeland Rao in 2015. They also showed that no such set exists forn ≤ 10.


SÉBASTIEN LABBÉ



1. Introduction


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The Domino Problem


SÉBASTIEN LABBÉ



1. Introduction


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SUBSTITUTIVE STRUCTURE OF JEANDEL-RAO APERIODIC TILINGS 3

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Figure 3. A finite part of a Jeandel-Rao aperiodic tiling in �0. Any tiling in the minimalsubshift X0 of �0 that we describe can be decomposed uniquely into 19 supertiles (two ofsize 45, six of size 72, four of size 70 and seven of size 112). The thick black lines show thecontour of the supertiles. The figure illustrates two complete supertiles of size 72 and 45.While Jeandel-Rao tile set T0 is not self-similar, the 19 supertiles corresponding to T12 areself-similar. The figure illustrates the presence of a horizontal fracture of 0’s which impliesthat 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 : �5 æ �4 is not onto. The problem comes from the existence of tilingsin �0, �1, �2, �3 and �4 that have horizontal fractures of 0’s or 1’s. The tilings in �4 obtainedby sliding along the fracture line can not be obtained as the image under the application of fi ofa tiling in �5. But we believe that fi is onto up to a set of measure 0. Thus if this conjectureholds, we give a complete characterization of the substitutive structure of tilings �0 made withJeandel-Rao’s tiles.

The article is structured as follows. In Section 2, we present the necessary definitions andnotations on Wang tiles including self-similarity, recognizability and aperiodicity. In Section 3, werecall the definition of markers, the desubstitution of Wang shifts and we propose algorithms tocompute them. In Section 4, we desubstitute tilings from �0 to �4. In Section 5, we show wecan remove two tiles from T Õ

4 to get T4 µ T Õ4 . In Section 6, we construct the tile set T5 by adding

decorations to tiles T4 to avoid fracture lines in tilings of �4 and we prove that fi : �5 æ �4 isan embedding. In Section 7, we construct the shearing topological conjugacy ÷ : �6 æ �5. InSection 8, we desubstitute tilings from �6 to �12. In Section 9, we prove the main results. InSection 10 we contruct eight tilings in �12 that are fixed by the square of Ê12. In Section 11, weshow that �4 \X4 is nonempty. In Section 12, we list in a table the tile sets Ti, markers Mi µ Ti

and morphisms �i+1 æ �i for every i with 0 Æ i Æ 12.

2. The search for small aperiodic tile sets

R. Penrose, Pentaplexity: A class of non-periodic tilings of theplane, Math. Intelligencer 2 (1979), 32–37.

Relaxing the rules

At this point, the search for other aperiodic sets of prototilesbegan.

Specifically, we allow tiles to be any shape, and we no longercount rotations.

In the 1970s Roger Penrose found a set of aperiodic prototileswith just two prototiles!

Interestingly, combining these tiles in various ways gives riseto a set of Wang tiles with only 24 tiles (which would havebeen the record at the time had anyone noticed). See Exercise11.1.2 in Grunbaum and 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 oversmall businesses or individuals, but when it comes to thepopulation of Great Britain being invited by a multi-national towipe their bottoms on what appears to be the work of a Knight ofthe Realm without his permission, then a last stand must bemade.” - David Bradley, director of Pentaplex (the company thatcares for Penrose’s copyrights)

The einstein (one stone) problem

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

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

not allow reflections of the monotile,have a monotile that is homeomorphic to a closed disc, andhave 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 isturning his attention to this problem... so who knows.

Further relaxations of the rules

One can ask for an einstein with less stringent rules. Here aresome that the experts have deemed reasonable. An einsteincould

allow reflections of the monotile,not be homeomorphic to a closed disc,have matching rules that are not forced by shape alone; forexample, colour matching rules with more than one colourallowed to meet, orhave matching rules that reach beyond adjacent tiles.

3. The Socolar–Taylor monotile

tiles must form continuous black stripes, and flag decorationsat opposite ends of each tile edge must point in the samedirection. (The arrows in (b) point to the twoflags at oppositeends of a vertical tile edge.) Each tile in (c) is a rotation and/orreflectionof the single prototile, and the only way to fill spacewhile obeying the rules everywhere is to form a nonperiodic,hierarchical extension of the pattern in (c).

Defining the einsteinTwo constructions that could conceivably be counted aseinsteins were discovered in 1995. A single prototile thatforces a pattern of the Penrose type was presented byGummelt (with a complementary proof by Steinhardt andJeong) [8, 9]. But in this case tiles are allowed to overlap andthe covering of the space is not uniform. For this reason theprototile is not considered to be an einstein.

The uniformly space-filling, three-dimensional prototileof Figure 2, a rhombic biprism, was exhibited by Schmitt,Conway, and Danzer [10]. To fill space, one is forced toconstruct 2D periodic layers of tiles sharing triangular faces,with ridges running in the direction of one pair of rhombusedges on top and the other pair below. The layers are thenstacked such that each is rotated by an angle / with respect tothe one below it, where / is the acute angle of the rhombicbase. Any choice of / other than integer multiples of p/3 orp/4 produces a tiling that is not periodic, and certain choicespermit a tiling in which the number of nearest neighborenvironments is finite, so that the prototile can be endowedwith bumps and nicks in a way that locks the relative posi-tions of adjacent layers.

Again, however, the universal reaction was ‘‘This is notreally what we are looking for.’’ The nonperiodicity of the

(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

tiling that respects the matching rules.

Figure 2. The SCD prototile and the space–filling tiling it forces.

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AU

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OR

S JOSHUA E. S. SOCOLAR received hisPh.D. in Physics from the University ofPennsylvania in 1987, with a thesis onquasilattices and quasicrystals. He has beenon the faculty at Duke since 1992, and he isaffiliated with the Center for Nonlinear andComplex Systems and the Duke Center forSystems Biology. His hobbies include jazzpiano, singing, and word games.

Physics Department and Center forNonlinear and Complex SystemsDuke UniversityDurham, NC 27708USAe-mail: [email protected]

JOAN M. TAYLOR took up mathematics in1991 at age 34 after being inspired by amagazine article on quasicrystals featuringPenrose’s rhombus tiling. She began but didnot complete a degree, preferring to conducther own research. Since then she has pursuedtiling and related topics in abstract algebra andnumber theory including original work onconstructible polygons. She likes to unwindwith knitting and reading.

Post Office Box U91Burnie, TAS 7320Australia

! 2011 Springer Science+Business Media, LLC, Volume 34, Number 1, 2012 19

tiles must form continuous black stripes, and flag decorationsat opposite ends of each tile edge must point in the samedirection. (The arrows in (b) point to the two flags at oppositeends of a vertical tile edge.) Each tile in (c) is a rotation and/orreflectionof the single prototile, and the only way to fill spacewhile obeying the rules everywhere is to form a nonperiodic,hierarchical extension of the pattern in (c).

Defining the einsteinTwo constructions that could conceivably be counted aseinsteins were discovered in 1995. A single prototile thatforces a pattern of the Penrose type was presented byGummelt (with a complementary proof by Steinhardt andJeong) [8, 9]. But in this case tiles are allowed to overlap andthe covering of the space is not uniform. For this reason theprototile is not considered to be an einstein.

The uniformly space-filling, three-dimensional prototileof Figure 2, a rhombic biprism, was exhibited by Schmitt,Conway, and Danzer [10]. To fill space, one is forced toconstruct 2D periodic layers of tiles sharing triangular faces,with ridges running in the direction of one pair of rhombusedges on top and the other pair below. The layers are thenstacked such that each is rotated by an angle / with respect tothe one below it, where / is the acute angle of the rhombicbase. Any choice of / other than integer multiples of p/3 orp/4 produces a tiling that is not periodic, and certain choicespermit a tiling in which the number of nearest neighborenvironments is finite, so that the prototile can be endowedwith bumps and nicks in a way that locks the relative posi-tions of adjacent layers.

Again, however, the universal reaction was ‘‘This is notreally what we are looking for.’’ The nonperiodicity of the

(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

tiling that respects the matching rules.

Figure 2. The SCD prototile and the space–filling tiling it forces.

.........................................................................................................................................................

AU

TH

OR

S JOSHUA E. S. SOCOLAR received hisPh.D. in Physics from the University ofPennsylvania in 1987, with a thesis onquasilattices and quasicrystals. He has beenon the faculty at Duke since 1992, and he isaffiliated with the Center for Nonlinear andComplex Systems and the Duke Center forSystems Biology. His hobbies include jazzpiano, singing, and word games.

Physics Department and Center forNonlinear and Complex SystemsDuke UniversityDurham, NC 27708USAe-mail: [email protected]

JOAN M. TAYLOR took up mathematics in1991 at age 34 after being inspired by amagazine article on quasicrystals featuringPenrose’s rhombus tiling. She began but didnot complete a degree, preferring to conducther own research. Since then she has pursuedtiling and related topics in abstract algebra andnumber theory including original work onconstructible polygons. She likes to unwindwith knitting and reading.

Post Office Box U91Burnie, TAS 7320Australia

! 2011 Springer Science+Business Media, LLC, Volume 34, Number 1, 2012 19

J.E.S Socolar and J.M. Taylor, An aperiodic hexagonal tile, J.Comb. Th. A 118 (2011), 2207–2231.

J.E.S Socolar and J.M. Taylor, Forcing nonperiodicity with a singletile, Math Intel. 34 (2012), 18–28.

J.M. Taylor, Aperiodicity of a functional monotile, preprint (2010);available fromhttp://www.math.uni-bielefeld.de/sfb701/preprints/view/420.

The Socolar–Taylor monotile

The existence of an aperiodic monotile was resolved almost adecade 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) Adjacenttiles must form continuous black stripes. Flag decorations at opposite ends of a tile edge (as indicated by the arrows) must pointin the same direction. (c) A portion of an infinite tiling. (For a color image, the reader is referred to the web version of thisarticle.)

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

1. The prototile

A version of the prototile, with its mirror image, is shown in Fig. 1. There are two constraints, 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 byPenrose [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 match-ing rules. The mirror symmetry of the tiles is not immediately apparent here; we have replacedleft-handed flags with blue stripes and right-handed with red. The matching rules are illustrated onthe right: R1 requires continuous black stripes across shared edges; and R2 requires the red or bluesegments at opposite endpoints of any given edge and collinear with that edge to be different col-ors. The white and gray tile colors are guides to the eye, highlighting the different reflections of theprototile.

Throughout this paper, the red and blue colors are assumed to be merely symbolic indicators ofthe 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 anintegral part of the reflection operation.

The matching rules presented in Figs. 1 and 2 would appear to be unenforceable by tile shapealone (i.e., without references to the colored decorations). One of the rules specifying how colorsmust match necessarily refers to tiles that are not in contact in the tiling and the other rule cannotbe implemented using only the shape of a single tile and its mirror image. Both of these obstaclescan be overcome, however, if one relaxes the restriction that the tile must be a topological disk. Fig. 3shows how the color-matching rules can be encoded in the shape of a single tile that consists ofseveral disconnected regions. In the figure, all regions of the same color are considered to composea single tile. The black stripe rule is enforced by the small rectangles along the tile edges. The red–blue rule is enforced by the pairs of larger rectangles located radially outward from each vertex. Theflag orientations (or red and blue stripe colors) are encoded in the chirality of these pairs. (For adiscussion of the use of a disconnected tile for forcing a periodic structure with a large unit cell,see [2].)

The following rules must hold in any given tiling:

R1 : the black curves must be continuous across all edges in thetiling, and

R2 : the flags at the vertices of two tiles separated by a single tileedge must always point in the same direction.

The Socolar–Taylor monotileThe matching rules on the Taylor–Socolar tile:

require reflections of the monotile andhave matching rules that reach beyond adjacent tiles.

However, if one is not concerned whether the monotile is atopological disc, then the matching rules can be forced byshape alone:

The hexagonal blocks on each arm have thickness h/3,allowing the blocks from three crossing arms to make a fullcolumn. The six arms on the prototile have outer faces thatare tilted from the vertical in a pattern that encodes the chi-rality of the flags of the 2D tile. Forming one triangular latticerequires that bevels of opposite type be joined, and hencethat flags of opposite chirality match in accordance with R2.

The small bumps on the tiles and the holes in the arms arearranged such that adjacent tiles can fit together if and only ifthe black stripes match up properly, as required by R1. Thethree square holes in each arm are positioned so that pro-jections from the faces on neighboring tiles can meet witheachother. Theholes are all the same; they donot themselvesencode the positions of the black stripes. Next, we create twotypes of plug that can be inserted into a hole. One type

consists of two square projections that fill opposite quadrantsof the hole; the other type fills the entire hole but only to halfits depth. The two types are both invariant under rotation by180!. Two plugs of the same type can fit together to fill a hole,but plugs of different types cannot. Finally, we place twocolumnsof threeplugs eachon eachof the large vertical facesof the main hexagonal portion of the tile. Each columnaligned with a black stripe has plugs of one type, and theother columns have plugs of the other type. (The latter areneeded to fill the holes in the arms at those positions.) Threeplugs are needed because of the staggered heights ofneighboring tiles. If a prototile that is a topological sphere isdesired, the plugs can be moved toward the middle of theirrespective faces so that the left and right side plugs meet andthe holes in the arms are converted to U-shaped slots.

Figure 5. The partial translational symmetry with the smallest spacing. Clusters of 24 shaded

tiles (two of each of the twelve tile orientations) are repeated throughout the tiling, forming a

triangular lattice. Purple stripes are shown only for a subset of one third of the tiles.

(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.

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Socolar–Taylor monotile

Theorem (Socolar–Taylor 2010)

The Socolar–Taylor monotile is aperiodic; that is, there are tilingssatisfying 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 13

7


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

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

We see that every possible cluster has two small cornersappearing together.


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 atile forces a black ring at the other vertex. (d) The forced periodic partial decoration. (For clearer visualization of the red andblue 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 theplane with no violation of the matching rules.

Proof. It is immediately clear that the hexagonal tiles (without colored markings) can fill space toform a triangular lattice of tiles. We begin with such a lattice of unmarked tiles and consider thepossibilities for adding marks consistent with R1 and R2. Note first that the configuration of darkblack 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 immediatelyforces the placement of a decoration (gray) that leads to the formation of a small ring. Thus the tilingmust 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 ringis 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 prototiledetermine the orientations of the red and blue diameters (but not the red–blue diameter). Given thecentral tile with a small black ring at its lower vertex, the vertical diameter of the tile at the lower leftmust be red. The tile at the upper left is then forced to be (partially) decorated as shown. R2 requiresthat the vertical stripe be blue. It cannot be just the bottom half of a red–blue diameter because thatwould not allow the black stripes to match. The vertical blue diameter forces the creation of anothersmall 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 additionalforced tiles, shows that the honeycomb lattice of small black rings and colored diameters shownin Fig. 4(d) is forced. The locations of the longer black stripes and the orientation of the red–bluediameter of each tile are not yet determined. We refer to the tiles that are partially decorated in thisfigure 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. 5must 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 neighborsof the added tile, the black stripes on the large tile have the same form as those on the L1 tiles andthus the large tiles obey R1. (Note that adding the long black stripe on the central tile will forcecorresponding long black stripes on the L1 tiles to its left and right.) Similarly, the forced orientationsof the red–blue diameters on the L1 next-nearest neighbors act to transfer R2 to the large tiles. Forthe large tiles, the matching of opposite colors is forced because each large tile edge is a red–blue

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


Now rule R2 is designed to ensure that if the corners of twoR1-triangles meet at a common tile, then they must have thesame length.

Since we have ensured that there must be at least one length0 triangle, R2 therefore implies that there must be ahexagonal grid of length 0 triangles:


An analogous argument shows that at least one length 1triangle exists. So R2 implies that we obtain a hexagonal gridof length 1 triangles.


Continuing this process ad infinitum gives a tiling of the planethat satisfies R1.


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

However, there is no largest size, so the resulting tiling mustnot have any translational periodicity; that is, the tiling isnonperiodic.


Fig. 1. The prototile and color matching rules. (a) The two tiles shown are related by reflection about a vertical line. (b) Adjacenttiles must form continuous black stripes. Flag decorations at opposite ends of a tile edge (as indicated by the arrows) must pointin the same direction. (c) A portion of an infinite tiling. (For a color image, the reader is referred to the web version of thisarticle.)

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

1. The prototile

A version of the prototile, with its mirror image, is shown in Fig. 1. There are two constraints, 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 byPenrose [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 match-ing rules. The mirror symmetry of the tiles is not immediately apparent here; we have replacedleft-handed flags with blue stripes and right-handed with red. The matching rules are illustrated onthe right: R1 requires continuous black stripes across shared edges; and R2 requires the red or bluesegments at opposite endpoints of any given edge and collinear with that edge to be different col-ors. The white and gray tile colors are guides to the eye, highlighting the different reflections of theprototile.

Throughout this paper, the red and blue colors are assumed to be merely symbolic indicators ofthe 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 anintegral part of the reflection operation.

The matching rules presented in Figs. 1 and 2 would appear to be unenforceable by tile shapealone (i.e., without references to the colored decorations). One of the rules specifying how colorsmust match necessarily refers to tiles that are not in contact in the tiling and the other rule cannotbe implemented using only the shape of a single tile and its mirror image. Both of these obstaclescan be overcome, however, if one relaxes the restriction that the tile must be a topological disk. Fig. 3shows how the color-matching rules can be encoded in the shape of a single tile that consists ofseveral disconnected regions. In the figure, all regions of the same color are considered to composea single tile. The black stripe rule is enforced by the small rectangles along the tile edges. The red–blue rule is enforced by the pairs of larger rectangles located radially outward from each vertex. Theflag orientations (or red and blue stripe colors) are encoded in the chirality of these pairs. (For adiscussion of the use of a disconnected tile for forcing a periodic structure with a large unit cell,see [2].)

4. A dendritic monotile

M. Mampusti and M. Whittaker, A monotile that forcesnonperiodicity through a local dendritic growth rule, preprint, toappear in Bull. LMS.

A dendritic monotile

Goal: Define a monotile that does not require reflections and eachtile only interacts with adjacent tiles.

Tilings must be constructed as follows. Once a single tile hasbeen placed, a direct isometry of the monotile can be addedto the plane provided the resulting collection of tiles is apatch, and

R1 the off-centre black lines and curves must be continuous acrosstiles (a la Socolar–Taylor) and

R2 the new tile’s red tree continuously connects with at least onered tree of an adjacent tile.


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

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

Therefore, a tiling T satisfies R2 if and only if the union ofR2-trees in T is connected.


The following two patches are legal:


Theorem (Mampusti-W, 2019)

Our dendritic monotile is aperiodic; that is, there are tilingssatisfying R1 and R2, and every such tiling is nonperiodic.

I will sketch the proof of existence, which follows fromconstructing an infinite nested union of patches.


P0P1 P2

P3

Constructing a tiling

Fix P0 at the origin, then we obtain a tiling of the plane:

T :=∞⋃n=1

Pn

5. An orientational monotile

J. Walton and M. Whittaker: An aperiodic tile with edge-to-edgeorientational matching rules, preprint.

An orientational monotileGoal: Find a monotile with similar properties to the Socolar-Taylormonotile, but with edge-to-edge matching rules.

Two tiles t1 and t2 are permitted to meet along a shared edgee only if:

R1 the off-centre black lines and curves must be continuous acrosse (a la Socolar–Taylor) and

R2 whenever the two charges at e in t1 and t2 both have aclockwise orientation then they must be opposite in charge.

The drawback is that we allow more flexibility with whatcounts as a ‘rule’ for edge matchings.

An orientational monotile

Theorem (Walton–W, 2019)

Our orientational monotile is aperiodic; that is, there are tilingssatisfying R1 and R2, and every such tiling is nonperiodic.

In this case, I want to sketch the proof that any tilingconstructed 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 R1triangles, and then assign R1 segments the same charge asthe 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.


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

E1 aN E2 ⇒ ch(E1) = ch(E2), E1 aF E2 ⇒ ch(E1) = (−1) · ch(E2)

This immediately implies that no chain E1 aF E2 aF E3 aF E1

of 3 edges.

Proof of nonperiodicitySo edges that meet ‘far’ must lead to longer edges, or else wecould create a spiral leading to a cycle of 3 edges.


So either:a) there must be R1 triangles of arbitrarily large size orb) 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 followingpattern of triangles above and below the line.

Aperiodic tilings: from the Domino problem to an aperiodic ...

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