Related papers: Fractal tiles associated with shift radix systems
Let $R$ be an $n\times n$ expanding matrix with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set $\mathcal{D}\subset\mathbb{Z}^n$ so that the integral self-affine set…
Let $\sigma$ be a non-unit Pisot substitution and let $\alpha$ be the associated Pisot number. It is known that one can associate certain fractal tiles, so-called \emph{Rauzy fractals}, with $\sigma$. In our setting, these fractals are…
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…
In this paper, we present high-level overviews of tile-based self-assembling systems capable of producing complex, infinite, aperiodic structures known as discrete self-similar fractals. Fractals have a variety of interesting mathematical…
In this paper, we study two classes of planar self-similar fractals $T_\varepsilon$ with a shifting parameter $\varepsilon$. The first one is a class of self-similar tiles by shifting $x$-coordinates of some digits. We give a detailed…
Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set…
Fractals and quasiperiodic structures share self-similarity as a structural property. Motivated by the link between Fibonacci fractals and quasicrystals which are scaled by the golden mean ratio $\frac{1+\sqrt{5}}{2}$, we introduce and…
Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show…
The pinwheel triangle of Conway and Radin is a standard example for tilings with self-similarity and statistical circular symmetry. Many modifications were constructed, all based on partitions of triangles or rectangles. The fractal example…
We set up a geometrical theory for the study of the dynamics of reducible Pisot substitutions. It is based on certain Rauzy fractals generated by duals of higher dimensional extensions of substitutions. We obtain under certain hypotheses…
We consider digit systems $(A,\mathcal{D})$, where $ A \in \mathbb{Q}^{n\times n}$ is an expanding matrix and the digit set $\mathcal{D}$ is a suitable subset of $\mathbb{Q}^n$. To such a system, we associate a self-affine set $\mathcal{F}…
We introduce a fractal version of the pinwheel substitution tiling. There are thirteen basic prototiles, all of which have fractal boundaries. These tiles, along with their reflections and rotations, create a tiling space which is mutually…
We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information about the connectedness of interior and boundary sets, and on…
Self-similar sets with open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples…
For a (non-unit) Pisot number $\beta$, several collections of tiles are associated with $\beta$-numeration. This includes an aperiodic and a periodic one made of Rauzy fractals, a periodic one induced by the natural extension of the…
This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be…
In this paper, we study arithmetical and topological properties for a class of Rauzy fractals ${\mathcal R}_a$ given by the polynomial $x^3- ax^2+x-1$ where $a \geq 2$ is an integer. In particular, we prove the number of neighbors of…
Fractal structures emerge from statistical and hierarchical processes in urban development or network evolution. In a class of efficient and robust geographical networks, we derive the size distribution of layered areas, and estimate the…
Although the theory of self-affine tiles and the theory of Rauzy fractals are quite different from each other, they have some common features. Both, self-affine tiles and Rauzy fractals have tiling properties and these tiling properties can…
In this paper, we propose to enumerate all different configurations belonging to a specific class of fractals: A binary initial tile is selected and a finite recursive tiling process is engaged to produce auto-similar binary patterns. For…