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A new simple construction of an aperiodic tile set based on self-referential (fixed point) argument is proposed. People often say about some discovery that it appeared “ahead of time”, meaning that it could be fully understood only in the context of ideas developed later. For the topic of this note, the construction of an aperiodic tile set based on the fixed-point (self-referential) approach, the situation is exactly the opposite. It should have been found in 1960s when the question about aperiodic tile sets was first asked: all the tools were quite standard and widely used at that time. However, the history had chosen a different path and many nice geometric ad hoc constructions were developed instead (by Berger, Robinson, Penrose, Ammann and many others, see [6]; a popular exposition of Robinson-style construction is given in [3]). In this note we try to correct this error and present a construction that should have been discovered first but seemed to be unnoticed for more that forty years. LIF Marseille, CNRS & University Aix-Marseille. Partially supported by ANR (Sycomore and Nafit grants) and RFBR (05-01-02803, 06-01-00122a), IITP RAS. !" #$%%"&'( )* &!" +, -. !" 1 The statement: aperiodic tile sets A tile is a square with colored sides. Given a set of tiles, we want to find a tiling, i.e., to cover the plane by (translated copies of) these tiles in such a way that colors match (a common side of two neighbor tiles has the same color in both). For example, if tile set consists of two tiles (one has black lower and left side 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 Figure 1: Tile set that has only periodic tilings and white right and top sides, the other has the opposite colors), it is easy to see that only periodic (checkerboard) tiling is possible. However, if we add some other tiles the resulting tile set may admit also non-periodic tilings (e.g., if we add all 16 possible tiles, any combination of edge colors becomes possible). It turns out that there are other tile set that have only aperiodic tilings. Formally: let C be a finite set of colors and let τ ⊂ C be a set of tiles; the components of the quadruple are interpreted as upper/right/lower/left colors of a tile. Our example tile set with two tiles is represented then as {〈white,white, black, black〉, 〈black, black,white,white〉}. A τ-tiling is a mapping Z → τ that satisfies matching conditions. Tiling U is called periodic if it has a period, i.e., if there exists a non-zero vector T ∈ Z such that U(x + T ) = U(x) for all x. Now we can formulate the result (first proven by Berger [1]): Tiles appeared first in the context of domino problem posed by Hao Wang. Here is the original formulation from [10]: “Assume we are given a finite set of square plates of the same size with edges colored, each in a different manner. Suppose further there are infinitely many copies of each plate (plate type). We are not permitted to rotate or reflect a plate. The question is to find an effective procedure by which we can decide, for each given finite set of plates, whether we can cover up the whole plane (or, equivalently, an infinite quadrant thereof) with copies of the plates subject to the restriction that adjoining edges must have the same color.” This question (domino problem) is closely related to the existence of aperiodic tile sets: (1) if they did not exist, domino problem would be decidable for some simple reasons (one may look in parallel for a periodic tiling or a finite region that cannot be tiled) and (2) the aperiodic tile sets are used in the proof of the undecidability of domino problem. However, in this note we concentrate on aperiodic tile sets only. !"#$% &' () !" "# $% $&'()*% !" Proposition. There exists a finite tile set τ such that τ-tilings exist but all of them are aperiodic. There is a useful reformulation of this result. Instead of tilings we can consider two-dimensional infinite words in some finite alphabet A (i.e., mappings of type Z 2 → A) and put some local constraints on them. This means that we choose some positive integer N and look at the word through a window of size N × N. Local constraint then says which patterns of size N × N are allowed to appear in a window. Now we can reformulate our Proposition as follows: there exists a local constraint that is consistent (some infinite words satisfy it) but implies aperiodicity (all satisfying words are aperiodic). It is easy to see that these two formulations are equivalent. Indeed, the color matching condition is 2 × 2 checkable. On the other hand, any local constraint can be expressed in terms of tiles and colors if we use N × N-patterns as tiles and (N −1)×N-patterns as colors; e.g., the right color of (N ×N)-tile is the tile except for its left column; if it matches the left color of the right neighbor, these two tiles overlap correctly. 2 Why theory of computation? At first glance this proposition has nothing to do with theory of computation. However, the question appeared in the context of the undecidability of some logical decision problems, and, as we shall see, can be solved using theory of computations. (A rare chance to convince “normal” mathematicians that theory of computations is useful!) The reason why theory of computation comes into play is that rules that determine the behavior of a computation device — say, a Turing machine with onedimensional tape — can be transformed into local constraints for the space-time diagram that represents computation process. So we can try to prove the proposition as follows: consider a Turing machine with a very complicated (and therefore aperiodic) behavior and translate its rules into local constraints; then any tiling represents a time-space diagram of a computation and therefore is aperiodic. However, this naïve approach does not work since local constraints are satisfied also at the places where no computation happens (in the regions that do not contain the head of a Turing machine) and therefore allow periodic configurations. So a more sophisticated approach is needed. !" #$%%"&'( )* &!" +, -. !" 3 Self-similarity The main idea of this more sophisticated approach is to construct a “self-similar” set of tiles. Informally speaking, this means that any tiling can be uniquely split by vertical and horizontal lines into M × M blocks that behave exactly like the individual tiles. Then, if we see a tiling and zoom out with scale 1 : M, we get a tiling with the same tile set. Let us give a formal definition. Assume that a non-empty set of tiles τ and positive integer M > 1 are fixed. A macro-tile is a square of size M × M filled with matching tiles from τ. Let ρ be a non-empty set of macro-tiles. Definition. We say that τ implements ρ if any τ-tiling can be uniquely split by horizontal and vertical lines into macro-tiles from ρ. Now we give two examples that illustrate this definition: one negative and one positive. Negative example: Consider a set τ that consists of one tile with all white sides. Then there is only one macro-tile (of given size M × M). Let ρ be a oneelement set that consists of this macro-tile. Any τ-tiling (i.e., the only possible τtiling) can be split into ρ-macro-tiles. However, the splitting lines are not unique, so τ does not implements ρ. Positive example: Let τ is a set of M tiles that are indexed by pairs of integers modulo M: The colors are pairs of integers modulo M arranged as shown