Related papers: Algorithms for translational tiling
Aperiodic tilings are non-periodic tilings defined by local rules. They are widely used to model quasicrystals, and a central question is to understand which of the non-periodic tilings are actually aperiodic. Among tilings, those by rhombi…
Tilings of a surface of negative Euler characteristic by n-gons with n\ge 7 is a finite problem. One extreme of the finite problem is single tile tilings. We develop the algorithm for finding all the single tile tilings and present the…
Three results are shown on producibility in the hierarchical model of tile self-assembly. It is shown that a simple greedy polynomial-time strategy decides whether an assembly A is producible. The algorithm can be optimized to use O(|A|…
The paper provides an elementary proof of Kenyon's necessary condition for the existence of a periodic tiling of the plane by squares with given periods. A similar new result on covering both sides of a rectangle by nonoverlaping squares is…
Our earlier article proved that if $n > 1$ translates of sublattices of $Z^d$ tile $Z^d$, and all the sublattices are Cartesian products of arithmetic progressions, then two of the tiles must be translates of each other. We re-prove this…
Let $T$ be a tile in $\mathbb{Z}^n$, meaning a finite subset of $\mathbb{Z}^n$. It may or may not tile $\mathbb{Z}^n$, in the sense of $\mathbb{Z}^n$ having a partition into copies of $T$. However, we prove that $T$ does tile $\mathbb{Z}^d$…
To a given tiling a non commutative space and the corresponding C*-algebra are constructed. This includes the definition of a topology on the groupoid induced by translations of the tiling. The algebra is also the algebra of observables for…
Classical results on aperiodic tilings are rather complicated and not widely understood. Below, an alternative approach is discussed in hope to provide additional intuition not apparent in classical works.
In this paper we study colorings (or tilings) of the two-dimensional grid $\mathbb{Z}^2$. A coloring is said to be valid with respect to a set $P$ of $n\times m$ rectangular patterns if all $n\times m$ sub-patterns of the coloring are in…
We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that theorem we show that every Ammann A2 tiling is self-similar in the sense of [B. Solomyak, Nonperiodicity implies unique composition for…
A subset V of GF(2)^n is a tile if GF(2)^n can be covered by disjoint translates of V. In other words, V is a tile if and only if there is a subset A of GF(2)^n such that V+A = GF(2)^n uniquely (i.e., v + a = v' + a' implies that v=v' and…
In this paper it is proved that there exist periodic monohedral tilings and finite seeds of colored tiles, which force non-periodic coloring of the whole plane
The periodic tiling conjecture (PTC) asserts, for a finitely generated Abelian group $G$ and a finite subset $F$ of $G$, that if there is a set $A$ that solves the tiling equation $\mathbb{1}_F * \mathbb{1}_A = 1$, there is also a periodic…
Is there a fixed dimension $n$ such that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by…
Kautz and de Bruijn graphs have a high degree of connectivity which makes them ideal candidates for massively parallel computer network topologies. In order to realize a practical computer architecture based on these graphs, it is useful to…
In this paper, we give a proof that it is undecidable whether a set of five polyominoes can tile the plane by translation. The proof involves a new method of labeling the edges of polyominoes, making it possible to assign whether two edges…
We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions,…
We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with O(n) Kolmogorov complexity of its n-by-n squares. We construct tile sets for which this…
By reformulating Wang tiles with tensors, we propose a natural generalization to the probabilistic and quantum setting. In this new framework, we introduce notions of tilings and periodicity directly extending their classical counterparts.…
In an earlier article [3], we presented an algorithm that can be used to rigorously check whether a specific cosine or sine polynomial is nonnegative in a given interval or not. The algorithm proves to be an indispensable tool in…