Related papers: Tiling the Euclidean and Hyperbolic planes with ri…

We obtain complete topological classification of states of nematic liquid crystal in the geometry of periodic array of rectangular posts between two parallel slabs, with tangent or normal boundary conditions. This classification has…

One well studied way to construct quasicrystalline tilings is via inflate-and-subdivide (a.k.a. substitution) rules. These produce self-similar tilings--the Penrose, octagonal, and pinwheel tilings are famous examples. We present a…

A general scheme for determining and studying integrable deformations of algebraic curves is presented. The method is illustrated with the analysis of the hyperelliptic case. An associated multi-Hamiltonian hierarchy of systems of…

We consider tilings of the plane with 12-fold symmetry obtained by the cut and projection method. We compute their cohomology groups using the techniques introduced by the second author, Hunton and Kellendonk. To do this we completely…

There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the…

Every body knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other…

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…

We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by…

We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with rational angles in degree: they are a one-parameter family of symmetric $a^4b$-pentagonal subdivisions of the tetrahedron with…

We propose a new example of discrete holography that provides a new step towards establishing the AdS/CFT duality for discrete spaces. A class of boundary Hamiltonians is obtained in a natural way from regular tilings of the hyperbolic…

We present a technique to lift some tilings of the discrete hyperbolic plane -- tilings defined by a 1D substitution -- into a zero entropy subshift of finite type (SFT) on non-abelian amenable Baumslag-Solitar groups $BS(1,n)$ for…

Here are two problems. First, understand the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describe the topology of connected components of plane sections of a centrally symmetric subsurface $S \subset…

We classify the dihedral edge-to-edge tilings of the sphere by regular polygons with gonality at least 5 and rhombi.

We develop the basic tools for classifying edge-to-edge tilings of the sphere by congruent pentagons. Then we prove that, for the edge combination $a^2b^2c$, such tilings are three two-parameter families of pentagonal subdivisions of the…

The Dedekind tessellation -- the regular tessellation of the upper half-plane by the Mobius action of the modular group -- is usually viewed as a system of ideal triangles. We change the focus from triangles to circles and give their…

We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the…

Brane tilings are efficient mnemonics for Lagrangians of N=2 Chern-Simons-matter theories. Such theories are conjectured to arise on M2-branes probing singular toric Calabi-Yau fourfolds. In this paper, a simple modification of the…

In this paper, we look at the improvement of our knowledge on a family of tilings of the hyperbolic plane which is brought in by the use of Sergeyev's numeral system based on grossone. It appears that the information we can get by using…

We develop a geometric framework for generalized Milnor classifying spaces in the setting of diffeological spaces and infinite-dimensional geometry. Starting from Milnor's construction, we introduce spherical and projective models endowed…

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a…