Related papers: Coordinates for a new triangular tiling of the hyp…
In this paper we study discreteness of complex hyperbolic triangle groups of type $[m,m,0;3,3,2]$, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $3,3,2$ in complex geodesics with…
Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an…
Inclusion properties are studied for balls of the triangular ratio metric, the hyperbolic metric, the $j^*$-metric, and the distance ratio metric defined in the unit ball domain. Several sharp results are proven and a conjecture about the…
We prove that acylindrically hyperbolic groups are monotileable. That is, every finite subset of the group is contained in a finite tile. This provides many new examples of monotileable groups, and progress on the question of whether every…
Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus…
In this paper, we study the contribution of the theory of grossone to the study of infinigons in the hyperbolic plane. We can see that the theory of grossone can help us to obtain much more classification for these objects than in the…
We present and prove closed form expressions for some families of binomial determinants with signed Kronecker deltas that are located along an arbitrary diagonal in the corresponding matrix. They count cyclically symmetric rhombus tilings…
Random tilings are interesting as idealizations of atomistic models of quasicrystals and for their connection to problems in combinatorics and algorithms. Of particular interest is the tiling entropy density, which measures the relation of…
We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a…
$\SLR$ geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from the 3-dimensional Lie group of all $2\times 2$ real matrices with determinant one. Our aim is to describe and visualize the {\it regular infinite…
Which polygons admit two (or more) distinct lattice tilings of the plane? We call such polygons double tiles. It is well-known that a lattice tiling is always combinatorially isomorphic either to a grid of squares or to a grid of regular…
We analytically study bond percolation on hyperbolic lattices obtained by tiling a hyperbolic plane with constant negative Gaussian curvature. The quantity of our main concern is $p_{c2}$, the value of occupation probability where a unique…
We apply the theory of Groebner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions T_N=T_{3k-1} and T_N=T_{3k} in a hexagonal…
On objects of a triangulated category with a stability condition, we construct a topology.
R. Nandakumar asked whether there is a tiling of the plane by pairwise incongruent triangles of equal area and equal perimeter. Recently a negative answer was given by Kupavskii, Pach and Tardos. Still one may ask for weaker versions of the…
Tiling models are classical statistical models in which different geometric shapes, the tiles, are packed together such that they cover space completely. In this paper we discuss a class of two-dimensional tiling models in which the tiles…
This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers basic tools in hyperbolic geometry and…
We describe a new infinite family of line arrangements in the projective plane with only triple points singularities and recover previously known examples.
We define a notion of tiling of the full infinite $p$-ary tree, establishing a series of equivalent criteria for a subtree to be a tile, each of a different nature; namely, geometric, algebraic, graph-theoretic, order-theoretic, and…
Colour symmetries with ten colours are presented for different tilings. In many cases, the existence of these colourings were predicted by group theoretical methods. Only in a few cases explicit constructions were known, sometimes using…