Class Of Fractals Research Articles

On the computation of some code sets of the added Sierpinski triangle

In recent years, the intrinsic metrics have been formulated on the classical fractals. In particular, Sierpinski-like triangles such as equilateral, isosceles, scalene, added and mod$-3$ Sierpinski triangle have been considered in many different studies. The intrinsic metrics can be defined in different ways. One of the methods applied to obtain the intrinsic metric formulas is to use the code representations of the points on these self-similar sets. To define the intrinsic metrics via the code representations of the points on fractals makes also possible to investigate different geometrical, topological properties and geodesics of these sets. In this paper, we investigate some circles and closed sets of the added Sierpinski triangle and express them as the code sets by using its intrinsic metric.

Approximation to the classical fractals by using non-affine contraction mappings

In the literature, there are various methods to obtain the fractal sets such as escape time algorithm, L-systems and iterated function system (IFS), etc. In this study, we aim to approximate to the classical fractals by using non-affine contraction mappings. In order to get these non-affine mappings, we utilize from the sequences of suitable Lipschitz continuous functions. Then, we obtain some approximations to the fractals which can be constructed as the attractor of an IFS. Finally, we give some illustrations for some specific cases.

Peano continua with self regenerating fractals

We deal with the question of Masayoshi Hata: is every Peano continuum a topological fractal? A compact space X is a topological fractal if there exists F a finite family of self-maps on X such that X=⋃f∈Ff(X) and for every open cover U of X there is n∈N such that for all maps f1,…,fn∈F the set f1∘…∘fn(X) is contained in some set U∈U.In the paper we present some idea how to extend a topological fractal and we show that a Peano continuum is a topological fractal if it contains so-called self regenerating fractal with nonempty interior. A Hausdorff topological space A is a self regenerating fractal if for every non-empty open subset U, A is a topological fractal for some family of maps constant on A∖U.The notion of self regenerating fractal much better reflects the intuitive perception of self-similarity. We present some classical fractals which are self regenerating.

On topological conjugacy of some chaotic dynamical systems on the Sierpinski gasket

The dynamical systems on the classical fractals can naturally be obtained with the help of their iterated function systems. In the recent years, different ways have been developed to define dynamical systems on the self similar sets. In this paper, we give composition functions by using expanding and folding mappings which generate the classical Sierpinski Gasket via the escape time algorithm. These functions also indicate dynamical systems on this fractal. We express the dynamical systems by using the code representations of the points. Then, we investigate whether these dynamical systems are topologically conjugate (equivalent) or not. Finally, we show that the dynamical systems are chaotic in the sense of Devaney and then we also compute and compare the periodic points.

The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations

To formulate the intrinsic metrics by using the code representations of the points on the classical fractals is an important research area since these formulas help to prove many geometrical and structural properties of these fractals. In various studies, the intrinsic metrics on the code set of the Sierpinski gasket, the Sierpinski tetrahedron, and the Vicsek (box) fractal are explicitly formulated. However, in the literature, there are not many works on the intrinsic metric that is obtained by the code representations of the points on fractals. Moreover, as seen in the studies on this subject, the contraction coefficients of the associated iterated function systems (IFSs) are the same for each fractal. In this paper, we define the intrinsic metric formula on the added Sierpinski triangle, whose IFS has different contraction factors, by using the code representations of the points of it. Finally, we give several geometrical properties of this fractal by using the intrinsic metric formula.

Minimally actuated hyper-redundant robots: Motion planning methods based on fractals and self-organizing systems

This article presents a motion planning method for a novel hyper-redundant robot with minimal actuation based on the principles of fractals and self-organizing systems. The robot consists of multiple links connected by passive joints and a movable actuator. The actuator travels over the links to a given joint and adjusts the relative angle between the two adjacent links allowing the robot to undergo the same wide range of motions of hyper-redundant robot but with only one actuator. A suitable objective of the motion planner is to minimize the number of actuator traversals, which translates into minimizing the number of bends in the c-space trajectory. To this end, we propose a novel method for motion planning using fractals and self-organizing systems. A self-similar pattern for the path is implemented to map a path from start to finish. Each iteration of path segments is of smaller dimension than the previous one and is appended to it, just as in classical fractals. This process continues until a feasible trajectory is calculated. Self-organizing systems are then applied to this trajectory post-processing to optimize it by eliminating bends in the path. Examples of the robot maneuvering around obstacles and through confined spaces are shown to demonstrate the efficacy of the motion planner.

EQUATIONS AND INTERVAL COMPUTATIONS FOR SOME FRACTALS

Very few characteristic functions, or equations, are reported so far for fractals. Such functions, called Rvachev functions in function-based modeling, are zero on the boundary, negative for inside points and positive for outside points. This paper proposes Rvachev functions for some classical fractals. These functions are convergent series, which are bounded with interval arithmetic and interval analysis in finite time. This permits to extend the Recursive Space Subdivision (RSS) method, which is classical in Computer Graphics (CG) and Interval Analysis, to fractal geometric sets. The newly proposed fractal functions can also be composed with classical Rvachev functions today routinely used in Constructive Solid Geometry (CSG) trees of CG or function-based modeling.

Affine Automata and Classical Fractals

This article reveals the relationship between fractal geometry and automata theory. It is shown that at least classical fractals are attractors of contracting affine automata and also that the symbolic space of a contracting affine automaton is continuously mapped onto its attractor.

Classical Fractals and Quantum Chaos in Ultracold Dipolar Collisions.

We examine a dipolar-gas model to address fundamental issues regarding the correspondence between classical chaos and quantum observations in ultracold dipolar collisions. The theoretical model consists of a short-range Lennard-Jones potential well with an anisotropic, long-range dipole-dipole interaction between two atoms. Both the classical and quantum dynamics are explored for the same Hamiltonian of the system. The classical chaotic scattering is revealed by the fractals developed in the scattering function (defined as the final atom separation as a function of initial conditions), while the quantum chaotic features lead to the repulsion of the eigenphases from the corresponding quantum S matrix. The nearest-eigenphase-spacing statistics have an intermediate behavior between the Poisson and the Wigner-Dyson distributions. The character of the distribution can be controlled by changing an effective Planck constant or the dipole moment. The degree of quantum chaos shows a good correspondence with the overall average of the classical scattering function. The results presented here also provide helpful insights for understanding the role of the inherent dipole-dipole interaction in the currently ongoing experiments on ultracold collisions of highly magnetic atoms.

“Graph paper” trace characterizations of functions of finite energy

We characterize functions of finite energy in the plane in terms of their traces on the lines that make up “graph paper” with squares of side length mn for all n and certain 1/2-order Sobolev norms on the graph paper lines. We also obtain analogous results for functions of finite energy on two classical fractals: the Sierpinski gasket and the Sierpinski carpet.

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.We examine the basic properties of this new notion of dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc.As our first application, we generalize the celebrated result of Chayes, Chayes and Durrett about the phase transition of the connectedness of the limit set of Mandelbrot's fractal percolation process. They proved that certain curves show up in the limit set when passing a critical probability, and we prove that actually ‘thick’ families of curves show up, where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large Hausdorff dimension. The proof of this is basically a lower estimate of the topological Hausdorff dimension of the limit set. For the sake of completeness, we also give an upper estimate and conclude that in the non-trivial cases the topological Hausdorff dimension is almost surely strictly below the Hausdorff dimension.Finally, as our second application, we show that the topological Hausdorff dimension is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic continuous function (in the sense of Baire category) defined on a compact metric space.

Topological conformal dimension

We investigate a quasisymmetrically invariant counterpart of the topological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension, gives a lower bound on the topological Hausdorff dimension of quasisymmetric images of the space. We obtain results concerning the behavior of this quantity under products and unions, and compute it for some classical fractals. The range of possible values of the topological conformal dimension is also considered, and we show that this quantity can be fractional.

ON THE VISUAL COMPLEXITY OF BUILT AND NATURAL LANDSCAPES

In this study, we analyze some critical points of the application of the box-counting method to the evaluation of the fractal dimensions of the natural and built landscapes. A brief theoretical discussion of the eventual drawbacks of the method is supported by experimental results of two box-counting programs applied to classical fractals. The optimized version of the algorithm, based on the results of computations for the classical fractal images, is proposed and employed for the evaluation of the complexity level of the chosen historical buildings and surrounding environment in the well-known case of Amasya city. The hypothesis of the relationship between the visual complexity of built and natural settings is analyzed for the Amasya case and for two historical Brazilian cities.

Powers of Distances to Lower Dimensional Sets as Muckenhoupt Weights

Let (X, d, μ) be an Ahlfors metric measure space. We give sufficient conditions on a closed set $${F \subseteqq X}$$ and on a real number β in such a way that d(x, F) β becomes a Muckenhoupt weight. We give also some illustrations to regularity of solutions of partial differential equations and regarding some classical fractals.

Normal Vector Based Subdivision Scheme to Generate Fractal Curves

In this paper, we firstly devise a new and general p-ary subdivision scheme based on normal vectors with multi-parameters to generate fractals. Rich and colorful fractals including some known fractals and a lot of unknown ones can be generated directly and conveniently by using it uniformly. The method is easy to use and effective in generating fractals since the values of the parameters and the directions of normal vectors can be designed freely to control the shape of generated fractals. Secondly, we illustrate the technique with some design results of fractal generation and the corresponding fractal examples from the point of view of visualization, including the classical Levy curves, Dragon curves, Sierpinski gasket, Koch curve, Koch-type curves and other fractals. Finally, some fractal properties of the limit of the presented subdivision scheme, including existence, self-similarity, non-rectifiability, and continuity but nowhere differentiability are described from the point of view of theoretical analysis. DOI: http://dx.doi.org/10.11591/telkomnika.v11i8.302 5

Fractal properties of MV-algebra pastings

Abstract We implement the fractal construction process to the MV-algebra pastings. Pasted MV-algebras generate special types of difference posets (D-posets). We compute the dimension of new fractal D-posets and show that the fractal structures discussed in this paper have a non-integer fractal dimension. Furthermore we observe an interesting agreement between the fractal dimension of the classical linear fractals (Cantor sets, Koch curves) and the dimension of fractal D-posets (Cantor fractal D-posets and Koch fractal D-posets).

Fractal analysis and its impact factors on pore structure of artificial cores based on the images obtained using magnetic resonance imaging

Pore structure is one of important factors affecting the properties of porous media , but it is difficult to describe the complexity of pore structure exactly. Fractal theory is an effective and available method for quantifying the complex and irregular pore structure. In this paper, the fractal dimension calculated by box-counting method based on fractal theory was applied to characterize the pore structure of artificial cores. The microstructure or pore distribution in the porous material was obtained using the nuclear magnetic resonance imaging (MRI). Three classical fractals and one sand packed bed model were selected as the experimental material to investigate the influence of box sizes, threshold value, and the image resolution when performing fractal analysis. To avoid the influence of box sizes, a sequence of divisors of the image was proposed and compared with other two algorithms (geometric sequence and arithmetic sequence) with its performance of partitioning the image completely and bringing the least fitted error. Threshold value selected manually and automatically showed that it plays an important role during the image binary processing and the minimum-error method can be used to obtain an appropriate or reasonable one. Images obtained under different pixel matrices in MRI were used to analyze the influence of image resolution. Higher image resolution can detect more quantity of pore structure and increase its irregularity. With benefits of those influence factors, fractal analysis on four kinds of artificial cores showed the fractal dimension can be used to distinguish the different kinds of artificial cores and the relationship between fractal dimension and porosity or permeability can be expressed by the model of D = a − b ln( x + c ). ► The proposed sequence of divisors in BC method can eliminate the border effect. ► Appropriate threshold value can be obtained using the minimum-error method. ► Higher image resolution brings more accurate estimation of fractal dimension. ► Fractal dimension can be used to distinguish the different kinds of artificial cores. ► Relationship between FD and porosity or permeability is expressed by D = a − b ln( x + c ).

Some results on the upper convex densities for the self-similar sets

In this paper, by means of a basic result concerning the estimation of the lower bounds of upper convex densities for the self-similar sets, we show that in the Sierpinski gasket, the minimum value of the upper convex densities is achieved at the vertices. In addition, we get new lower bounds of upper convex densities for the famous classical fractals such as the Koch curve, the Sierpinski gasket and the Cartesian product of the middle third Cantor set with itself, etc. One of the main results improves corresponding result in the relevant reference. The method presented in this paper is different from that in the work by Z. Zhou and L. Feng [The minimum of the upper convex density of the product of the Cantor set with itself, Nonlinear Anal. 68 (2008) 3439–3444].

Hausdorff measure of uniform self-similar fractals

Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R d. Denote by D the Hausdorff dimension, by H D(E) the Hausdorff measure and by diam(E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H D(E) = diam(E)D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present a parametrised UIFS, where ω depends on c and show the inequatily H D(E) < diam(E)D, if c is small enough.

Completeness of Muckenhoupt classes

In this note we prove that the Hausdorff distance between compact sets and the Kantorovich distance between measures, provide an adequate setting for the convergence of Muckenhoupt weights. The results which we prove on compact metric spaces with finite metric dimension can be applied to classical fractals.