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. This paper is the first of four papers, which together seek a complete characterization of the triples (ABC,N, T ) such that ABC can be N-tiled by T . In this paper, we consider...

The objective of the study was to compare three methods of monitoring the inhibition by dietary theaflavins of N-nitrosomethylbenzylamine (NMBA)-induced rat esophageal intraepithelial neoplasia: the mean tile grade, measured by computer-assisted quantitative image tile analysis; tumor multiplicity; and mean tumor size. A "tile" is defined as a small portion of a microscopic image at x 40, 87 x ...

We look at sets of tiles that can tile any region of size greater than 1 on the square grid. This is not the typical tiling question, but relates closely to it and therefore can help solve other tiling problems – we give an example of this. We also present a result to a more classic tiling question with dominoes and L-shape tiles.

In this paper, we improve the approach of the previous paper about the domino problem in the hyperbolic plane[8, 9]. This time, we prove that the general problem of tiling the hyperbolic plane withà la Wang tiles is undecidable.

We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not necessarily local) transformations called flips. This study allows us to formulate an exhaustive generation algorithm and a uniform random sampling algorithm. We fina...

In this paper, we prove that the general tiling problem of the hyperbolic plane is undecidable by proving a slightly stronger version using only a regular polygon as the basic shape of the tiles. The problem was raised by a paper of Raphael Robinson in 1971, in his famous simplified proof that the general tiling problem is undecidable for the Euclidean plane, initially proved by Robert Berger i...

We study the error robustness of tilings of the plane. The fundamental question is the following: given a tileset, what happens if we allow a small probability of errors? Are the objects we obtain close to an error-free tiling of the plane? We prove that tilesets that produce only periodic tilings are robust to errors. For this proof, we use a hierarchical construction of islands of errors (see...

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). Generating stochastic wall patterns is crucial in the creation of a wide range of backgrounds (castles, temples, ruins...). A specificWang tile set was introduced previously to tackle this problem, in a ...