Related papers: Fractal dual substitution tilings
Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…
A general construction principle of inflation rules for decagonal quasiperiodic tilings is proposed. The prototiles are confined to be polygons with unit edges. An inflation rule for a tiling is the combination of an expansion and a…
We develop a multifractal random tilling that fills the square. The multifractal is formed by an arrangement of rectangular blocks of different sizes, areas and number of neighbors. The overall feature of the tilling is an heterogeneous and…
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…
Fractal geometries, characterized by self-similar patterns and non-integer dimensions, provide an intriguing platform for exploring topological phases of matter. In this work, we introduce a theoretical framework that leverages isospectral…
Fractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the…
In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to…
We compute the Cech cohomology with integer coefficients of one-dimensional tiling spaces arising from not just one, but several different substitutions, all acting on the same set of tiles. These calculations involve the introduction of a…
The primary objectives of this paper are to present the construction of bivariate fractal interpolation functions over triangular interpolating domain using the concept of vertex coloring and to propose a double integration formula for the…
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…
This is a chapter in an upcoming book on aperiodic order. We go over different versions of tiling cohomology (\v Cech, pattern-equivariant, PV, quotient) with emphasis on the inverse limit constructions used to compute these cohomologies.…
A new method for constructing aperiodic tilings is presented. The method is illustrated by constructing a particular tiling and its hull. The properties of this tiling and the hull are studied. In particular it is shown that these tilings…
In this paper we study self-similar and fractal networks from the combinatorial perspective. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to…
Recent advances in deep learning have transformed many fields by introducing generic embedding spaces, capable of achieving great predictive performance with minimal labeling effort. The geology field has not yet met such success. In this…
Accurate segmentation of long and thin tubular structures is required in a wide variety of areas such as biology, medicine, and remote sensing. The complex topology and geometry of such structures often pose significant technical…
We propose a formalism for tilings with infinite local complexity (ILC), and especially fusion tilings with ILC. We allow an infinite variety of tile types but require that the space of possible tile types be compact. Examples include…
This paper gives new solutions to the problem: 'Can we construct monohedral tilings of the disk such that a neighbourhood of the origin has trivial intersection with at least one tile?'
Given an n-dimensional substitution whose associated linear expansion is unimodular and hyperbolic, we use elements of the one-dimensional integer \v{C}ech cohomology of the associated tiling space to construct a finite-to-one…
The term fractal describes a class of complex structures exhibiting self-similarity across different scales. Fractal patterns can be created by using various techniques such as finite subdivision rules and iterated function systems. In this…
In the article "Construction of the continuous hull for the combinatorics of a regular pentagonal tiling of the plane" we constructed the continuous hull for the combinatorics of "A regular pentagonal tiling of the plane", and in the…