Related papers: Tiling of Hyperbolic Surface by a Single Tile

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…

A tiling is a decomposition of a polygon into finitely many non-overlapping triangles. We prove that if a regular n-gon, $n \geq 5$, $n \neq 28$, can be tiled with similar right triangles, then one of the angles of these triangles is in…

We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed…

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,…

We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of…

An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the…

We show that the problem of tiling the Euclidean plane with a finite set of polygons (up to translation) boils down to prove the existence of zeros of a non-negative convex function defined on a finite-dimensional simplex. This function is…

An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer $k\geq 3$, there exists a normal tiling of the Euclidean plane by convex…

We describe a method to classify crystallographic tilings of the Euclidean and hyperbolic planes by tiles whose stabiliser group contains translation isometries or whose topology is not that of a closed disk. We tackle this problem from two…

We develop the necessary machinery in order to prove that hexagonal tilings are uniquely determined by their Tutte polynomial, showing as an example how to apply this technique to the toroidal hexagonal tiling.

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}…

We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides.

We show how to determine if a given simple rectilinear polygon can be tiled with rectangles, each having an integer side.

We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than $\pi$) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with…

In this paper, we consider the finite tiling problem which was proved undecidable in the Euclidean plane by Jarkko Kari in 1994. Here, we prove that the same problem for the hyperbolic plane is also undecidable.

In 1946 Fine and Niven posed problem E724, asking to demonstrate that every hypercube can be tiled by any number of hypercubic tiles larger than some value. This requires only basic number theory, but the problem of finding the smallest…

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…

Roughly, a conformal tiling of a Riemann surface is a tiling where each tile is a suitable conformal image of a Euclidean regular polygon. In 1997, Bowers and Stephenson constructed an edge-to-edge conformal tiling of the complex plane…

We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via…

This paper studies properties of tilings of the plane by parallelograms. In particular it is established that in parallelogram tilings using a finite number of shapes all tiles occur in only finitely many orientations.