Aperiodic tilings with one prototile and low complexity atlas matching rules
We give a constructive method that can decrease the number of prototiles needed to tile a space. We achieve this by exchanging edge to edge matching rules for a small atlas of permitted patches. This method is illustrated with Wang tiles, and we apply our method to present via these rules a single prototile that can only tile aperiodically, and a pair of square tiles that can only tile aperiodically.
The field of aperiodic tilings was created by Berger’s discovery  of a set of square tiles which were strongly aperiodic in the sense that they could only tile the plane in a non-repeating global structure. Unsurprisingly there has been some interest on how far this number could be decreased. The number of tiles was reduced over time, to square tiles by Robinson  in , and, relaxing to non-square tiles, to tiles by Penrose  two years later. This led naturally to serious consideration about the possible existence of a single tile. While a simple example has not been forthcoming, if we relax the requirement that the monotile be completely defined by its shape alone, there has been progress.
In  Socolar studied a more general problem, ’-isohedral’ monotiles, which had the monotile as a limiting case. Relaxing conditions on edge-coloring, non-connected tiles or space-filling provided positive results, but not in the limit. In 1996 Gummelt  considered tiles that where allowed to overlap, and produced a decorated tile which could force strong aperiodicity.
This paper extends work by Goodman-Strauss on ‘atlas matching rules’. In  Goodman-Strauss describes how by requiring a tiling be covered by a suitable finite atlas of permitted bounded configurations, a domino can serve as a monotile. Sadly the atlas requires patches of extremely large radius. This paper describes a method of altering matching rules from coloured tiles to atlas matching rules with very small patches. Furthermore if two of the tiles have the same shape, the number of prototiles needed is decreased.
We use this method to construct a pair of square tiles which tile aperiodically, and a single cubic tile that tiles aperiodically.
2 The atlas matching rule construction
We shall describe the basic definitions we will be using in this paper, drawing many of them from . For clarity we will be limiting the spaces we are tiling to for some . With minor alterations the method will work in any homogeneous space (for example hyperbolic space ).
Let be a finite set of compact subsets of , each the closure of its interior. Denote these subsets as prototiles. Let be a group of isometries of , which includes all translations of . The groups we will be using most in this paper are the group of translations and the group of all isometries . For a given set of prototiles and a group of isometries , define a tile as the image , for some , . A patch for is a set of tiles with pairwise disjoint interiors and the support of a patch is the union of its tiles. A tiling with prototiles is a patch with support . We shall refer to the support of a prototile as .
We now want to introduce the notion of ‘decorating’ a prototile, and hence all tiles produced from it. Construct a function where is a set containing a distinguished element, say, and possibly other elements. A point in the prototile is c-coloured if . We will refer to points that are -coloured as uncoloured points.
Extend to points of any given tile by for each .
A coloured tiling satisfies the identical facet (matching) rule if for all tiles , (where ), for all .
This covers cases where two tiles ‘match’ if they have the same colour on the interior of their shared boundary (for example Wang tiles). We will be using a slightly more general version of this rule in the rest of this paper, which allows tiles to match under wider conditions, as follows.
An facet (matching) rule is a function such that and .
A coloured tiling satisfies the facet (matching) rule if for all tiles , (where ),
for all .
We describe below a way of translating from this style of matching rule to the following matching rule.
A tiling satisfies an atlas (matching) rule if there exists an atlas of patches such that for every tile , there exists a patch about (with being in the strict interior of ) such that is a translation of some .
Furthermore must have a finite number of elements, and any patch in must be compact.
A prototile set satisfies the atlas (matching) rule if all tilings with prototiles satisfy the atlas matching rule .
In this paper, we will be using patches defined by the ‘1-corona’ about a tile . The ‘1-corona’ of a tile is the set of tiles touching (see ).
A tiling is a -tiling if it has a prototile set with allowable isometries and colouring , and satisfies the facet rule .
A tiling is a -tiling if it has a prototile set with allowable isometries , and satisfies the atlas matching rule .
Two tilings are MLD (mutually locally derivable) if one is obtained from the other in a unique way by local rules, and vice versa.
A -tiling is MLD to a -tiling for some -corona atlas rule and a prototile set with .
Take and partition it into a set of equivalence classes , where iff up to the action of an element of . For all , let be the largest subgroup of such that for all and all , .
Enumerate the elements of as .
Choose the smallest you can so as to construct an injective function . Define .
We now have a construction taking prototiles to ordered pairs of a prototile from and an automorphism of that prototile. Observe that is a subset of .
Proof of Theorem 1
Define a new prototile set , where is as just defined. Let the set of allowable functions from the prototiles into be , instead of . Take the set of allowable -coronas in the -tiling , and replace every tile originating from a translation of a prototile with , with and originating from . This will give you a set of -corona patches of . Use this set as the atlas rule for .
has facet rules, which are intrinsic to the set of allowable first coronas (since the set of allowable first coronas list what boundaries are allowed to meet each other). Since our definition of and its atlas correspond to the first coronas of tiles in , with replaced by , any tiling by is MLD to a tiling from . Since then .
Take a prototile set and partition it into a set of equivalence classes , as in the previous construction. If there exists such that , there exists a prototile set (with atlas rules) which tiles with less prototiles than .
We know that , thus . ∎
This method of construction produces a prototile set with cardinality .
is strongly aperiodic iff is strongly aperiodic. (See  for definition). This is because every tiling in is MLD to a tiling in , and strong aperiodicity is preserved under MLD equivalency.
3 Motivating examples and further improvements
For a simple illustration of the method, let us consider a tiling of the plane by 13 Wang tiles (unit squares with matching rules defined by matching coloured edges) as given in [3, 4]. Label the Wang tiles as . We can apply the above construction to get a function from to , where is the group of symmetries of the square.
For example, enumerate the symmetries of the square as . Then such a function could send to , and the remaining tiles to . The result is shown in diagram 1, for a small patch of the tiling.
As is common with Wang tiles, the colouring of is represented as actual colours superimposed onto the tile. We represent the change of prototile set from to by adding a colouring to and , which looks like their alphabetical symbols. This colouring has uncoloured points on the exterior of the tile (and thus no effect on the matching rules), but admits a free action by the symmetry group of a square.
Consider Kari’s Wang cube prototiles, . This is a set of 21 unit cube prototiles, with facet matching rules that only tile aperiodically.
Choose a unit cube prototile which has an asymmetric label; i.e., for any two distinct isometries of the cube , .
Since the set of isometries of the cube is of cardinality 48, we can choose 21 unique isometries . We use the method in Construction to replace with .
Thus we have an aperiodic protoset with one prototile which is MLD to Kari’s Wang Cubes. Note that we have lost the property of matching rules being determined on faces, and replaced them with a set of legal one corona patches (which cannot be rotated or reflected, of course). We have also had to broaden the set of allowable mappings of the prototiles into the tiled space, from translations to translations and rotation/reflections.
This algorithm can be further improved, by partitioning into equivalence classes based on what prototiles have the same support up to isometry, not just translation.
Let be a -tiling as in Construction . If there is a prototile whose support is a non-trivial isometry of another prototile (where ), then the resulting -tiling may have less prototiles than one originating from Construction .
Partition , where iff up to the action of an element of .
Further partition , where iff up to the action of an element of .
This two-stage partitioning gives us a collection of equivalence classes () as per the first construction. Additionally we know that there exist isometries in from elements of to elements of . Take the with the largest cardinality and denote it . From the definition of there exists an isometry such that . Furthermore we know that an given isometry can only take elements from one set to (by definition of equivalence class). Thus we can replace any prototile in with a unique isometry of a prototile in , since .
By applying the previous construction to , we can get a minimal uncoloured prototile set that can be used to translate prototiles in , and hence , to atlas rules.
Take a prototile set of equilateral triangles, as shown in figure 2. The prototiles have two different orientations, and three (could be up to six) colours.
You partition into , since all prototiles in have the same support, up to isometry. You then further partition , where is the set of prototiles with point upwards, and is the set of prototiles with point downwards. Denote the first prototile of as . Applying the first construction to gives you , for , and . While this is sufficient to define the tiling, it has the problem that any picture of the tiling needs to include information about the isometries used for each tile. Thus we replace with a tile with an uncoloured boundary, but with a coloured interior which is not preserved under any non-identity element of .
Thanks are due to Chaim Goodman-Strauss, Joshua E. S. Socolar and Edmund Harriss for helpful conversations on this paper. My supervisor John Hunton has assisted considerably with improving the readability of this paper. We also thank the University of Leicester and EPSRC for a doctoral fellowship. The results of this article will form part of the author’s PhD thesis.
- ‘Penrose tilings as coverings of congruent decagons’, Petra Gummelt, Geometriae Dedicata Volume 62, Number 1 / August, 1996. doi:10.1007/BF00239998.
- ‘Open Questions in Tilings’, Chaim Goodman-Strauss January 10, 2000 http://comp.uark.edu/ strauss/papers/survey.pdf
- ‘An aperiodic set of 13 Wang tiles’, Karel Culik II, Discrete Mathematics, Volume 160, Issues 1-3, 15 November 1996, Pages 245-251
- ‘A small aperiodic set of Wang tiles’, Jarkko Kari, Discrete Mathematics, Volume 160, Issues 1-3, 15 November 1996, Pages 259-264
- ‘An aperiodic set of Wang cubes’, Jarkko Kari, Karel Culik, Journal of Universal Computer Science, vol. 1, no. 10 (1995), 675-686
- ‘Cohomology of Substitution Tiling Spaces’, Marcy Barge, Beverly Diamond, John Hunton, Lorenzo Sadun, Ergodic Theory and Dynamical Systems, doi:10.1017/S0143385709000777
- ‘The undecidability of the domino problem’, Berger, R., Memoirs Amer. Math. Soc. 66(1966)
- ‘Proving theorems by pattern recognition II’, Wang, Hao (January 1961), Bell System Tech. Journal 40(1):141.
- ‘More ways to tile with only one shape polygon’, Joshua E. S. Socolar, The Mathematical Intelligencer, Volume 29, Number 2 / June, 2007, doi:10.1007/BF02986203
- ‘Undecidability and Nonperiodicity for Tilings of the Plane’, R. M. Robinson, Inventiones Mathematicae, 12(3), 1971 pp. 177209.
- ‘Role of aesthetics in pure and applied research’, Penrose, Roger (1974), Bulletin of the Institute of Mathematics and its Applications 10