Related papers: Fractal dual substitution tilings
This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The…
We develop the theory of fractal homeomorphisms generated from pairs of overlapping affine iterated function systems.
The fractal structure of spin clusters and their boundaries in the critical two-dimensional (2D) Ising model is investigated numerically. The fractal dimensions of these geometrical objects are estimated by means of Monte Carlo simulations…
A fractal is in essence a hierarchy with cascade structure, which can be described with a set of exponential functions. From these exponential functions, a set of power laws indicative of scaling can be derived. Hierarchy structure and…
The recently discovered Hat tiling admits a 4-dimensional family of shape deformations, including the 1-parameter family already known to yield alternate monotiles. The continuous hulls resulting from these tilings are all topologically…
A fractal surface is a set which is a graph of a bivariate continuous function. In the construction of fractal surfaces using IFS, vertical scaling factors in IFS are important one which characterizes a fractal feature of surfaces…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of…
The dynamics of frictional interfaces play an important role in many physical systems spanning a broad range of scales. It is well-known that frictional interfaces separating two dissimilar materials couple interfacial slip and normal…
In a region R consisting of unit squares, a (domino) tiling is a collection of dominoes (the union of two adjacent squares) which pave fully the region. The flip graph of R is defined on the set of all tilings of R where two tilings are…
This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using…
In this article, we present a novel box-covering algorithm for analyzing the fractal properties of complex networks. Unlike traditional algorithms that impose a predetermined box size, our approach assigns nodes to boxes identified by their…
We introduce a generalization of conventional lattice gauge theory to describe fracton topological phases, which are characterized by immobile, point-like topological excitations, and sub-extensive topological degeneracy. We demonstrate a…
We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. The predicted equations are derived in this model, and new equations can be discovered as well.…
We study the space of all tilings which can be obtained using the Robinson tiles (this is a two-dimensional subshift of finite type). We prove that it has a unique minimal subshift, and describe it by means of a substitution. This…
We demonstrate existence of a tile assembly system that self-assembles the statistically self-similar Sierpinski Triangle in the Winfree-Rothemund Tile Assembly Model. This appears to be the first paper that considers self-assembly of a…
It is broadly known that any parallelepiped tiles space by translating copies of itself along its edges. In earlier work relating to higher-dimensional sandpile groups, the second author discovered a novel construction which fragments the…
We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence…
Urbanization is a phenomenon of concern for planning and public health: projections are difficult because of policy changes and natural events, and indicators are multiple. There are previous studies of development that used fractals, but…
Many models of fractal growth patterns (like Diffusion Limited Aggregation and Dielectric Breakdown Models) combine complex geometry with randomness; this double difficulty is a stumbling block to their elucidation. In this paper we…