Related papers: Periodicity and ergodicity in the trihexagonal til…
We study the tiling of a two-dimensional region of the plane by $K$-cell one-dimensional tiles, or $K$-mers. Unlike previous studies, which typically allowed for one single value of $K$ or sometimes a small assortment of fixed values, here…
We classify the dihedral edge-to-edge tilings of the sphere by regular polygons with gonality at least 5 and rhombi.
Thin surfaces are ubiquitous in nature, from leaves to cell membranes, and in technology, from paper to corrugated containers. Structural thinness imbues them with flexibility, the ability to easily bend under light loads, even as their…
Tilings are around us everywhere, and our curiosity draws us to study their properties. A tiling is a way of arranging pieces on a board, such that there is no space left uncovered, nor any space covered by more than one tile. In…
In this paper we define an infinite family of triangular tilings of the hyperbolic plane defined by two parameters ranging in the natural nummbers and we give a uniform way to define coordinates for locating the triangles of the tiling.
A case study of arithmetic dynamics over the rationals on the Markoff surface is presented, in particular the local-global dynamical property of strong residual periodicity. The dynamical system induced by the composition of any two of the…
We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles.…
The paper demonstrates the geometric optic properties of the transition radiation (TR) of an ultrarelativistic point-like charged particle crossing the transverse wall of a circular semi-infinite waveguide with ideally conducting walls. It…
The article considers the optically thick accretion discs with the progressive increasing height. The surface is assumed to be the conical form. The radiation with considered wavelength emerges from an ring on the cone and is described by…
We consider a class of discontinuous piecewise linear differential systems in $\mathbb{R}^3$ with two pieces separated by a plane. In this class we show that there exist differential systems having: a unique limit cycle, a unique…
We experimentally study the mixing of binary granular systems in a horizontal rotating cylinder. When materials have the same size and differ by dynamic angle of repose only, we observe an axial transport of matter that generates transient…
The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular $k$-gonal tile…
Amorphous solids are mechanically rigid while possessing a disordered structure similar to that of dense liquids. Recent research indicates that dynamical heterogeneity, spatio-temporal fluctuations in local dynamical behavior, might help…
Given a collection of N rectangles such that the side ratio of each one is a quadratic irrationality, we find all rectangles which can be tiled by rectangles similar to one of the given ones. It means that each possible shape can be used…
We classify the dihedral edge-to-edge tilings of the sphere by squares and rhombi.
Given a tiling $\mathcal{T}$ of the plane by straight edge polygons, which is invariant by two independent translations, we construct a family of embedded triply periodic minimal surfaces which desingularizes $\mathcal{T}\times\mathbb{R}$.…
The geometric representation at a fixed frequency of the wavevector (or dispersion) surface $\omega(\vec k)$ for lossless, homogeneous dielectric--magnetic uniaxial materials is explored, when the elements of the relative permittivity and…
We report on recent developments in the dynamics and rigidity of infinite-volume homogeneous spaces, viewed through the lens of circles. By addressing four natural questions about circle packings, we highlight the interplay between…
Two-component mixtures in optical lattices reveal a rich variety of different phases. We employ an exact diagonalization method to obtain the relevant correlation functions in hexagonal optical lattices to characterize those phases. We…
We examine the optical properties of metals containing a periodic arrangement of nonoverlapping spherical mesopores, empty or filled with a dielectric material. We show that a slab of such a porous metal transmits light over regions of…