arXiv research

Study on tilings of the plane with two types of tiles of varying areas.

Study replicability in high-dimensional statistics, resolving open problems.

We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and say that a polyomino is crystallized if it has $h$ holes and $g(h)$ tiles. We analyze structural properties of crystall…

This note characterizes monohedral tilings of regular polygons with up to three tiles.

The study of tiling homology on flat surfaces, proving impossibility of certain tilings.

New tiles allow efficient knot mosaics for small knots.

Rep-tiles fill cubes in any dimension.

Study tiling spaces over irrational tori using diffeological classification.

Shellable tilings on simplicial complexes help understand their structure.

In this article we study Ammann tilings from the perspective of symplectic geometry. Ammann tilings are nonperiodic tilings that are related to quasicrystals with icosahedral symmetry. We associate to each Ammann tiling two explicitly constructed highly singular symplectic spaces and we show that they are diffeomorphic…

The study classifies tilings of the sphere by congruent quadrilaterals.

New method constructs tilings of the plane using directed edges and alignments.

New tile types for knots and links reduce complexity.

New spectral sequences derived from shellable tilings.

The main goal of this paper is to define a 1-1 correspondence between between substitution tilings constructed by inflation and the arithmetic of positional representation in the underlying real vector space. It introduces a generalization of inflationary tessellations to equivalence classes of tiles. Two tiles belong …

We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})…

We determine the topology of the moduli space of periodic tilings of the plane by parallelograms. To each such tiling, we associate combinatorial data via the zone curves of the tiling. We show that all tilings with the same combinatorial data form an open subset in a suitable Euclidean space that is homotopy equivalen…

Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.

Softens tilings in 3D space, proving conjectures about polyhedral tilings.

Paper proves corner connection tiles can represent knots with fewer tiles.

This paper classifies all 3D rep-tiles up to homeomorphism.

In this note we prove that any monohedral tiling of the closed circular unit disc with $k \leq 3$ topological discs as tiles has a $k$-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain t…

4-ball can be tiled with knotted surfaces.

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 different viewpoints, one with constructive techniques to enumerate such tiling…

The study explores maps of 2- and 3-uniform tilings on the torus.

Study higher rank inner products and their tilings to describe tori degenerations.

In this paper we describe the pentagonal tiling of the plane defined in the article "A regular pentagonal tiling of the plane" by P. L. Bowers and K. Stephenson as a conformal substitution tiling and summarize many of its properties given in the mentioned article. We show furthermore why such tiling is not FLC with res…

In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to …

The paper explores different perspectives on rhombile tilings.

Shear moves connect square-tiled surfaces in quadratic differentials.

The study finds arithmetic groups often in square-tiled surface monodromies.

This note connects tiling billiards dynamics to Novikov's problem via helicoidal construction.

Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to three-dimensional cube tilings. The connections between discrete extremal length …

Constructs surfaces that can be tiled by a finite set of rigid motion congruence classes of tiles.

A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these for…

We discuss the art and science of producing conformally correct euclidean and hyperbolic tilings of compact surfaces. As an example, we present a tiling of the Chmutov surface by hyperbolic (2, 4, 6) triangles.

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 Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To a…

The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this paper, we classify the simplest case, which is the edge-to-edge tilings of the 2-d…

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 that there is no primitive substitutive tiling of the hyperbolic plane $\mathbb{H…

In this paper we introduce the concept of a space-efficient knot mosaic. That is, we seek to determine how to create knot mosaics using the least number of non-blank tiles necessary to depict the knot. This least number is called the tile number of the knot. We determine strict bounds for the tile number of a knot in t…

Paper bridges matching rules and height functions in aperiodic tilings.

New method extracts hidden phases in binary mixtures using tubular tilings.

Investigates proving geometric theorems over complex and real numbers using tilings.

We call "flippable tilings" of a constant curvature surface a tiling by "black" and "white" faces, so that each edge is adjacent to two black and two white faces (one of each on each side), the black face is forward on the right side and backward on the left side, and it is possible to "flip" the tiling by pushing all …

This paper presents a novel scheme, based on a unique combination of genetic algorithms (GAs) and deep learning (DL), for the automatic reconstruction of Portuguese tile panels, a challenging real-world variant of the jigsaw puzzle problem (JPP) with important national heritage implications. Specifically, we introduce …

Paper introduces spherical knot mosaics for knot and link invariants.

New tiles in higher dimensions are shown to be homeomorphic to balls.

Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with colored cubes and bricks of $(2\times2\times n)$-board in three dimensions. After a short introduction and the definition of breakability we show a way …