Local Rules and Global Order, or Aperiodic Tilings Bruno Durand, Leonid Levin and Alexander Shen Can local rules impose a global order? If yes, when and how? This is a rather philosophical ques- tion that could be asked in many cases. How the local interaction of atoms creates crystals? (Or quasicrys- tals.) How one living cell manages to develop into a (say) pine cone whose seeds form spirals (and the number of spirals usually is a Fibonacci number)? Is it possible to program locally connected computers in such a way that the network is still functional if a small fraction of the nodes is corrupted? Is it possi- ble for a big team of people (or ants) each trying to reach her/his private goals, to behave reasonably? These questions range from theology to “politi- cal science” and are rather difficult. In mathematics the most prominent example of this kind is the so- called Berger theorem on aperiodic tilings (see be- low the exact statement). It was proved by Berger in 1966 [1]. 1 In 1971 the proof was simplified by Robinson [7] who invented the well known “Robin- son tiles” that could tile the entire plane but only in an aperiodic way (Fig. 1). Figure 1: Robinson tiles [reflections and rotations are allowed] Since then many similar constructions were invented (see, e.g., [3, 6]); some other proofs were based on different ideas (e.g. [4]). However, we did not man- age to find a publication which provides a short but complete proof of the theorem: Robinson tiles look simple but when you start to analyze them, you have to deal with many technical details. “This argument 1 In fact, the motivation at that time was related to the unde- cidability of some specific class of first order formulas, see [2] is a bit long and is not used in the remainder of the text, so it could be skipped on first reading” (says C. Radin in [6] about the proof). It’s a pity, however, toskip the proof of a nice the- orem whose statement can be understood by a high school student (unlike Fermat Theorem, you even don’t need to know anything about exponentiation). We try to fill this gap and provide a simple construc- tion of an aperiodic tiling with a complete proof, making the argument as simple as possible (increas- ing the number of tiles when necessary). Of course, simplicity is a matter of taste, so we can only hope you will find this argument simple and nice. If not, you can look at an alternative approach in [5]. Definitions Let A be a finite (nonempty) alphabet. A configura- tion is an infinite cell paper where each cell is occu- pied by a letter from A; formally, the configuration is a mapping of type Z 2 → A.A local rule is an ar- bitrary subset L ⊂ A 4 whose elements are considered as 2 × 2 squares: 〈a 1 , a 2 , a 3 , a 4 〉∈ L is a square a 1 a 2 a 3 a 4 We say that these squares are allowed by rule L. A configuration τ satisfies local rule L if all 2 × 2 squares in it are allowed by L. Formally it means that 〈τ (i, j + 1), τ (i + 1, j + 1), τ (i, j), τ (i + 1, j)〉∈ L for any i, j ∈ Z. A non-zero integer vector t =(t 1 , t 2 ) is a period of τ if t -shift preserves τ , i.e., τ (x 1 + t 1 , x 2 + t 2 )= τ (x 1 , x 2 ) for any x 1 , x 2 ∈ Z. Aperiodic Tilings theorem Theorem (Berger) There exist an alphabet A and a local rule L such that: 1