Tiling with bars under tomographic constraints

Discrete Applied Mathematics, 2009

On square or hexagonal lattices tiles or polyominoes are coded by words. The polyominoes that tile the plane by translation are characterized by the Beauquier-Nivat condition. By using the constant time algorithms for computing the longest common extensions in two words, we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of Gambini and Vuillon. For pseudo-hexagon polyominoes not containing arbitrarily large square factors we also have a linear algorithm. The results are extended to more general tiles.

2009

In this paper we study algorithms for tiling problems. We show that the conditions (T 1) and (T 2) of Coven and Meyerowitz [5], conjectured to be necessary and sufficient for a finite set A to tile the integers, can be checked in time polynomial in diam(A). We also give heuristic algorithms to find all non-periodic tilings of a cyclic group ZN. In particular we carry out a full classification of all non-periodic tilings of Z144.

Algorithmica, 2012

Discrete & Computational Geometry, 2017

We study the computational hardness of the tiling puzzle with polyominoes, where a polyomino is a rectilinear polygon (i.e., a polygon made by connecting unit squares.) In the tiling problem, we are given a rectilinear polygon P and a set S of polyominoes, and asked whether P can be covered without any overlap using translated copies of polyominoes in S. In this paper, we focus on trominoes and bars as polyominoes; a tromino is a polyomino consisting of three unit squares, and a bar is a rectangle of either height one or width one. Notice that there are essentially two shapes of trominoes, that is, Ishape (i.e., a bar) and L-shape. We consider the tiling problem when restricted to only L-shape trominoes, only I-shape trominoes, both L-shape and I-shape trominoes, or only two bars. In this paper, we prove that the tiling problem

Contributions Discret. Math., 2020

We present a new mathematical model for tiling finite subsets of $\mathbb{Z}^2$ using an arbitrary, but finite, collection of polyominoes. Unlike previous approaches that employ backtracking and other refinements of `brute-force' techniques, our method is based on a systematic algebraic approach, leading in most cases to an underdetermined system of linear equations to solve. The resulting linear system is a binary linear programming problem, which can be solved via direct solution techniques, or using well-known optimization routines. We illustrate our model with some numerical examples computed in MATLAB. Users can download, edit, and run the codes from http://people.sc.fsu.edu/~jburkardt/m_src/polyominoes/polyominoes.html. For larger problems we solve the resulting binary linear programming problem with an optimization package such as CPLEX, GUROBI, or SCIP, before plotting solutions in MATLAB.

Computing Research Repository, 2009

Discrete tomography deals with reconstructing finite spatial objects from lower dimensional projections and has applications for example in timetable design. In this paper we consider the problem of reconstructing a tile packing from its row and column projections. It consists of disjoint copies of a fixed tile, all contained in some rectangular grid. The projections tell how many cells are covered by a tile in each row and column. How difficult is it to construct a tile packing satisfying given projections? It was known to be solvable by a greedy algorithm for bars (tiles of width or height 1), and NP-hardness results were known for some specific tiles. This paper shows that the problem is NP-hard whenever the tile is not a bar.

The American Mathematical Monthly

The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting, and to provide a novel method for the construction of such polygonal tilings.

2012

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T , overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. We wish to understand possible tilings by completely characterizing the triples (ABC, T, N) such that ABC can be N-tiled by T. In particular, this understanding should enable us to specify for which N there exists a tile T and a triangle ABC that is N-tiled by T ; or given N , to determine which tiles and triangles can be used for N-tilings; or given ABC, to determine which tiles and N can be used to N-tile ABC. This is the fifth paper in a series of papers on this subject. In [1], we dealt with the case when ABC is similar to T , and the case when T is a right triangle. In [2], we proved a number of nonexistence theorems, leaving open two cases, one of which is dealt with in [3]. In [4], we took up the last remaining case: when ABC is not similar to T , and T has one angle equal to 120 • , and T is not isosceles (although ABC can be isosceles or even equilateral). Under those hypotheses, there are no known N-tilings. In [4], we showed that if there are any such tilings, then the tile is similar to one with integer side lengths. That is the problem we take up in this paper. We are still not able to completely solve the problem, but we prove that if there are any N-tilings by such tiles, then N ≥ 96. Combining this results with our earlier work, we can remove the exception for a 120 • tile, obtaining definitive non-existence results. For example, there is no 7-tiling, no 11-tiling, no 14-tiling, no 19-tiling, no 31-tiling, no 41-tiling, etc. Regarding the number N = 96: There are several possible shapes of ABC, and for each shape, we exhibit the smallest N for which it is presently unknown whether there is an N-tiling. For example, for equilateral ABC, the simplest unsolved case as of May, 2012 is N = 135. For each of these minimal-N examples, the tile would have to have sides (3, 5, 7).

Cornell University - arXiv, 2022

Ammann bars are formed by segments (decorations) on the tiles of a tiling such that forming straight lines with them while tiling forces nonperiodicity. Only a few cases are known, starting with Robert Ammann's observations on Penrose tilings, but there is no general explanation or construction. In this article we propose a general method for cut and project tilings based on the notion of subperiods and we illustrate it with an aperiodic set of 36 decorated prototiles related to what we called Cyrenaic tilings.

Azim Premji University, 2014