Related papers: A linear algorithm for Brick Wang tiling
The game and movie industries always face the challenge of reproducing materials. This problem is tackled by combining illumination models and various textures (painted or procedural patterns). Gnerating stochastic wall patterns is crucial…
Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging…
Deciding if a given set of Wang tiles admits a tiling of the plane is decidable if the number of Wang tiles (or the number of colors) is bounded, for a trivial reason, as there are only finitely many such tilesets. We prove however that the…
One of the most fundamental problems in tiling theory is the domino problem: given a set of tiles and tiling rules, decide if there exists a way to tile the plane using copies of tiles and following their rules. The problem is known to be…
This paper presents a stochastic Wang tiling based technique to compress or reconstruct disordered microstructures on the basis of given spatial statistics. Unlike the existing approaches based on a single unit cell, it utilizes a finite…
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,…
The paper presents a concept/technique to compress and synthesize complex material morphologies that is based on Wang tilings. Specifically, a microstructure is stored in a set of Wang tiles and its reconstruction is performed by means of a…
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.…
This investigation studies the decidability problem of plane edge coloring with three symbols. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges that have one of $p$ colors are arranged side by side such that…
We introduce a partial decidability protocol for the Wang tiling problem (which is the prototype of undecidable problems in combinatorics and statistical physics) by constructing a suitable mapping from tilings of finite squares of…
The computational complexity of tiling finite simply connected regions with a fixed set of tiles is studied in this paper. We show that the problem of tiling simply connected regions with a fixed set of $23$ Wang tiles is NP-complete. As a…
Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…
We introduce the idea that the P vs NP problem can have a finer structure. Given the NP complete problem of interest, the configurations space of the problem can be divided in (at least) two regions. In one region, polynomial algorithms to…
In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory. We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and…
The problem of rectangle tiling binary arrays is defined as follows. Given an $n \times n$ array $A$ of zeros and ones and a natural number $p$, our task is to partition $A$ into at most $p$ rectangular tiles, so that the maximal weight of…
This paper presents an algorithm for computing the contraction of two-dimensional tensor networks on a square lattice; and we combine it with solving congruence equations to compute the exact enumeration (including weighted enumeration) of…
In this article, we apply the techniques developed in our previous article ``Local generation of tilings'', in which we introduced two definitions capturing the intuitive idea that some subshifts admit a procedure that can generate any…
Motivated by the study of Fibonacci-like Wang shifts, we define a numeration system for $\mathbb{Z}$ and $\mathbb{Z}^2$ based on the binary alphabet $\{0,1\}$. We introduce a set of 16 Wang tiles that admits a valid tiling of the plane…
Wang tile based representation of a heterogeneous material facilitates fast synthesis of non-periodic microstructure realizations. In this paper, we apply the tiling approach in numerical homogenization to determine the Representative…
This paper presents an approach to constructing microstructural enrichment functions to local fields in non-periodic heterogeneous materials with applications in Partition of Unity and Hybrid Finite Element schemes. It is based on a concept…