Related papers: The finite tiling problem is undecidable in the hy…
In this paper, we consider the periodic tiling problem which was proved undecidable in the Euclidean plane by Yu. Gurevich and I. Koriakov in 1972. Here, we prove that the same problem for the hyperbolic plane is also undecidable.
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…
In this paper, we prove that the general problem of tiling the hyperbolic plane with \`a la Wang tiles is undecidable.
The present paper is a new version of the arXiv paper revisiting the proof given in a previous paper of the author published in 2008 proving that the general tiling problem of the hyperbolic plane is undecidable by proving a slightly…
In this paper we improve the approach of a previous paper about the domino problem in the hyperbolic plane, see arXiv.cs.CG/0603093. This time, we prove that the general problem of the hyperbolic plane with \`a la Wang tiles is undecidable.
In this paper, we complete the construction of paper arXiv:cs.CG/0701096v2. Together with the proof contained in arXiv:cs.CG/0701096v2, this paper definitely proves that the general problem of tiling the hyperbolic plane with {\it \`a la}…
The translational tiling problem, dated back to Wang's domino problem in the 1960s, is one of the most representative undecidable problems in the field of discrete geometry and combinatorics. Ollinger initiated the study of the…
We show that the following problem is undecidable: given two polygonal prototiles, determine whether the plane can be tiled with rotated and translated copies of them. This improves a result of Demaine and Langerman [SoCG 2025], who showed…
We provide a definitive classification of all finite sets of regular polygons that admit a tiling of the hyperbolic plane, thereby establishing the decidability of the Domino Problem for this class of prototiles. We show that admissibility…
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…
Does a given a set of polyominoes tile some rectangle? We show that this problem is undecidable. In a different direction, we also consider tiling a cofinite subset of the plane. The tileability is undecidable for many variants of this…
Altenbernd, Thomas and W\"ohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the B\"uchi and Muller ones [1]. It was proved…
We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove…
We prove that the following problem is co-RE-complete and thus undecidable: given three simple polygons, is there a tiling of the plane where every tile is an isometry of one of the three polygons (either allowing or forbidding…
We provide a resolution of the Heesch problem for homogeneous (also known as semi-regular) tilings, and as a corollary, for tilings by convex monotiles in the hyperbolic plane. We also provide the first known example of weakly aperiodic…
Translational tiling problems are among the most fundamental and representative undecidable problems in all fields of mathematics. Greenfeld and Tao obtained two remarkable results on the undecidability of translational tiling in recent…
To study the fixed parameter undecidability of tiling problem for a set of Wang tiles, Jeandel and Rolin show that the tiling problem for a set of 44 Wang bars is undecidable. In this paper, we improve their result by proving that whether a…
We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that this variant is undecidable regardless of the shapes,…
The finiteness problem for automaton groups and semigroups has been widely studied, several partial positive results are known. However we prove that, in the most general case, the problem is undecidable. We study the case of automaton…
In the 60's, Berger famously showed that translational tilings of $\mathbb{Z}^2$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings by translations of a…