Sierpinski Gasket Research Articles

Dirichlet fractional Gaussian fields on the Sierpinski gasket and their discrete graph approximations

We define and study the Dirichlet fractional Gaussian fields on the Sierpinski gasket and show that they are limits of fractional discrete Gaussian fields defined on the sequence of canonical approximating graphs.

Tunability of spin-wave spectra in a 2D triangular shaped magnonic fractals

Reprogramming the structure of the magnonic bands during their operation is important for controlling spin waves in magnonic devices. Here, we report the tunability of the spin-wave spectra for a triangular shaped deterministic magnonic fractal, which is known as Sierpinski triangle by solving the Landau–Lifshitz–Gilbert equation using a micromagnetic simulations. The spin-wave dynamics change significantly with the variation of iteration number. A wide frequency gap is observed for a structure with an iteration number exceeding some value and a plenty of mini-frequency bandgaps at structures with high iteration number. The frequency gap could be controlled by varying the strength of the magnetic field. A sixfold symmetry in the frequency gap is observed with the variation of the azimuthal angle of the external magnetic field. The spatial distributions of the spin-wave modes allow to identify the bands surrounding the gap. The observations are important for the application of magnetic fractals as a reconfigurable aperiodic magnonic crystals.

Ultra wideband (UWB) dielectric resonator antenna using fractal‐inspired feeding mechanism

SummaryThis article examines a compact dielectric resonator antenna (DRA) for UWB applications. Here a composite feeding structure excites a dielectric resonator which in tern provides the resonant modes TE111, TE121, TE212, and TE222. The dielectric resonator (DR), built of alumina ceramic (εr, DR = 9.8), is mounted on a fractal triangular patch, inspired by Sierpinski Gasket. The suggested DRA is supported by a FR4 substrate (εr, sub = 4.4) and measures compactly 40 × 30 × 8.5 mm3. To confirm the results of its simulations, the proposed antenna's prototype is prepared and measured up to the second iteration. The measured outcome demonstrates that the suggested antenna has a frequency range of 3.38–10.71 GHz (104%) for S11 < −10 dB and provides a maximum gain of 7.23 dBi at 8 GHz along with highest possible simulated efficiency of 98%. The suggested DRA is appropriate for small‐range future 4G/5G UWB wireless multimedia applications due to its ultrawide bandwidth, excellent gain, reasonable efficiency, omnidirectional radiation features, and compact construction. This DRA is also suitable for repeaters of mobile and vehicular communications.

Properties of Laughlin states on fractal lattices

Laughlin states have recently been constructed on fractal lattices and have been shown to be topological in such systems. Some of their properties are, however, quite different from the two-dimensional case. On the Sierpinski triangle, for instance, the entanglement entropy shows oscillations as a function of particle number and does not obey the area law despite being topologically ordered, and the particle density is non-uniform in the bulk. Here, we investigate these deviant properties in greater detail on the Sierpinski triangle, and we also study the properties on the Sierpinski carpet and the T-fractal. We find that the density variations across the fractal are present for all the considered fractal lattices and for most choices of the number of particles. The size of anyons inserted into the lattice Laughlin state also varies with position on the fractal lattice. We observe that quasiholes and quasiparticles have similar sizes and that the size of the anyons typically increases with decreasing Hausdorff dimension. As opposed to periodic lattices in two dimensions, the Sierpinski triangle and carpet have inner edges. We construct trial states for both inner and outer edge states. We find that oscillations of the entropy as a function of particle number are present for the T-fractal, but not for the Sierpinski carpet. Finally, we observe deviations from the area law for several different bipartitions on the Sierpinski triangle.

A New Fractal Topologies Based on Hypercube Interconnection Network

A parallel processing system's most crucial part is a network interconnection that links its processors. The hypercube topology has exciting features, making it an excellent option for parallel processing applications. This paper presents two innovative configurations of interconnection networks based on fractal Sierpinski and a hypercube. These are the Sierpinski Triangle Topology (STT) and Sierpinski Carpet Topology (SCT). Compared to a hypercube, the Sierpinski Triangle topology (STT) noticed a significant decrease in the number of nodes and links as large networks grew. Hence, it considers a great way to reduce costs because it uses fewer nodes and links. The average distance is also shorter, which is better. Despite it has a smaller bisection width and a more degree than a hypercube by one. The Sierpinski Carpet Topology (SCT) has the advantage of having a higher bisection width than a hypercube. That is preferable because it places a lower restriction on the difficulty of parallel algorithms. In contrast, the drawback of this topology is that it has a diameter and average distance more than a hypercube.

The trinomial triangle knitted shawl

We investigate a variation on Pascal's triangle and approximations to Sierpinski's triangle, by considering the coefficients in the trinomial expansion . These trinomial coefficients have many properties similar to those of the binomial coefficients. We illustrate the triangle of numbers with a knitted shawl.

Brownian motion on the golden ratio Sierpinski gasket

We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without a finitely ramified cell structure, via a study on the trace of an electrical network on an infinite graph. The Dirichlet form is the unique one that is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. The proof is based on a fixed point problem of a renormalization map, inspired by Sabot's celebrated work for finitely ramified fractals. Lastly, the Hunt process associated with the Dirichlet form satisfies a two-sided sub-Gaussian heat kernel estimate.

Comparison of some dynamical systems on the quotient space of the Sierpinski tetrahedron

In this paper, it is aimed to construct two different dynamical systems on the Sierpinski tetrahedron. To this end, we consider the dynamical systems on a quotient space of $\{ 0,1,2,3 \}^{\mathbb{N}}$ by using the code representations of the points on the Sierpinski tetrahedron. Finally, we compare the periodic points to investigate topological conjugacy of these dynamical systems and we conclude that they are not topologically equivalent.

Structure transformation from Sierpiński triangles to chains assisted by gas molecules.

Reversible transformations between fractals and periodic structures are of fundamental importance for understanding the formation mechanism of fractals. Currently, it is still a challenge to controllably achieve such a transformation. We investigate the effect of CO and CO2 molecules on Sierpiński triangles (STs) assembled from Fe atoms and 4,4″-dicyano-1,1':3',1″-terphenyl (C3PC) molecules on Au surfaces. Using scanning tunneling microscopy, we discover that the gas molecules induce a transition from STs into 1D chains. Based on density functional theory modeling, we propose that the atomistic mechanism involves the transformation of a stable 3-fold coordination Fe(C3PC)3 motif to Fe(C3PC)4 with an axially bonded CO molecule. CO2 causes the structural transformation through a molecular catassembly process.

Analytical and dimensional properties of fractal interpolation functions on the Sierpiński gasket

In this article, we construct the fractal interpolation functions (FIFs) on the Sierpiński gasket (SG) by taking data set at the nth level. We discuss the oscillation space, $$\mathcal {L}_q$$ space and some other function spaces on SG, and determine some conditions under which, this FIF belongs to these spaces. We obtain the fractal dimension of the graph of the FIF and the Hausdorff dimension of the invariant measure supported on the graph of the FIF. We also prove that this FIF has finite energy. After that, we discuss the restrictions of the FIF on the bottom line segment of SG and prove that the restriction is again an FIF on the bottom line. We estimate the fractal dimension of the graph of the Riemann-Liouville fractional integral of the restrictions of the FIF on the bottom line segment of SG. Lastly, we prove that the Hausdorff and box-counting dimension of the graph of the restriction of any harmonic function on the bottom line segment of SG are 1 and for the Riemann- Liouville fractional integral of this restriction function, these dimensions are also 1.

Reconstruction wedges in AdS/CFT with boundary fractallike structures

In this work, we show the robustness of uberholography and its associated quantum error correcting code against the breakdown of entanglement wedge in the presence of highly entropic mixed states in the bulk. We show that for Cantor-set-like erasure in the boundary in $AdS_3/CFT_2$, the code distance is independent of the mixed-state entropy in the bulk in the $m\rightarrow\infty$ limit. We also show that for a Sierpinski triangle shaped boundary subregion with fractal boundary erasures in $AdS_4/CFT_3$, bulk reconstruction is possible in the presence of highly entropic mixed states in the bulk in the large $m$ regime.

Topological Flatband Loop States in Fractal‐Like Photonic Lattices

AbstractNoncontractible loop states (NLSs) are a recently realized topological entity in flatband lattices, arising typically from the band touching at a point where a flat band intersects one or more dispersive bands. There exists also band touching across a plane, where one flat band overlaps another all over the Brillouin zone without crossing a dispersive band. Such isolated plane‐touching flat bands remain largely unexplored. For example, what are the topological features associated with such flatband degeneracy? Here, nontrivial NLSs and robust boundary modes in a system with such degeneracy are demonstrated. Based on a tailored photonic lattice constructed from the well‐known fractal Sierpinski gasket, the wavefunction singularities and the conditions for the existence of the NLSs are theoretically analyzed. It is shown that the NLSs can exist in both singular and nonsingular flat bands, as a direct reflection of the real‐space topology. Experimentally, directly such flatband NLSs in a laser‐written Corbino‐shaped fractal‐like lattice are observed. This work not only leads to a deep understanding of the mechanism behind the nontrivial flatband states, but also opens up new avenues to explore fundamental phenomena arising from the interplay of flatband degeneracy, fractal structures, and band topology.

Non-linear operators of divergence form on the Sierpinski gasket

We consider non-linear operators L of divergence type on the Sierpinski gasket. We prove the Harnack inequality for non-negative L-harmonic functions on the Sierpinski gasket. The proof is based on Moser’s method, and we construct a new measure λ by combining a Hausdorff measure and an energy measure, which plays an essential role at key steps. The Harnack inequality is achieved by involving the Caccioppoli type inequality and the weighted Sobolev inequality which give the local boundedness and the Weak Harnack inequality of solutions with respect to the measure λ.

The trapping problem on horizontal partitioned level-3 sierpinski gasket networks

Random walk on complex networks is a research hotspot nowadays. The average trapping time (ATT) is an important property related to the trapping problem, which is a variant of random walk, because it can be used to measure the transmission efficiency of particles randomly walking on the network. In this paper, we consider the trapping problem on the horizontal partitioned level-3 Sierpinski gasket network which is determined by the cutting line l k , that is, by the partition coefficient k. Then through the structure of this research model, we derive the exact analytical expression of the ATT. Furthermore, we draw two kinds of numerical simulation diagrams to simulate the relationship between the ATT and the iteration number and the partition coefficient, and compare them with the ATT on the original graph (uncut). The obtained solution shows that the ATT is affected by the k, specifically, the larger the k, the shorter the ATT, that is the higher the transmission efficiency.

CPW Fractal Antenna with Third Iteration of Pentagonal Sierpinski Gasket Island for 3.5 GHz WiMAX and 5.2 GHz WLAN Applications

Nowadays, the compact and multiband antennas are typically required for personal communication devices in the continuously developing wireless communication industry. The fractal antenna with the third iteration of the pentagonal Sierpinski gasket island for WiMAX and WLAN applications is presented in this work. It starts with the fundamental design of the square patch (Antenna A1) and pentagonal patch (Antenna A2). This simulation work is done using CST Microwave Studio simulation studio by applying the concept of the zero, first, second and third fractal iteration. Then, it goes on to use the fractal geometry concept of the Sierpinski gasket island structure with three designs step. The designs consist of the first iteration (Antenna B1), second iteration (Antenna B2) and third iteration (Antenna B3) of fractal geometry. The simulation work of Antenna B3 is compared with the fabrication work of the same design. After that, the measurement of the Antenna B3 is done in laboratory with - 29.55 dB at 3.41 GHz and - 20.40 dB at 5.28 GHz for its operating frequencies with bandwidth of 3.52 GHz and 5.48 GHz, respectively. At targeting 3.5 GHz WiMAX, 5.2 GHz WLAN application and 7.24 GHz of Antenna B3, the antenna shows the – 17.78 dB, - 29.63 dB and – 22.73 dB, respectively, and this value is feasible for WiMAX and WLAN operation.

Binary Apollonian networks.

There is a well-known relationship between the binary Pascal's triangle and the Sierpinski triangle, in which the latter is obtained from the former by successive modulo 2 additions starting from a corner. Inspired by that, we define a binary Apollonian network and obtain two structures featuring a kind of dendritic growth. They are found to inherit the small-world and scale-free properties from the original network but display no clustering. Other key network properties are explored as well. Our results reveal that the structure contained in the Apollonian network may be employed to model an even wider class of real-world systems.

Mechanical properties of 3D-printed hierarchical structures based on Sierpinski triangles

Incorporating structural hierarchy in the design of engineering composites can be used to tailor or enhance the physical and mechanical properties for specific functionalities. In this study, we propose a design of hierarchical triangular composite (HTC) composed of multi-scale Sierpinski triangles and fabricated using 3D printing with thermoplastic urethanes (TPU) material. A comprehensive framework incorporating theoretical and experimental methods is developed to study the effects of hierarchy order on the quasi-static, dynamic properties and energy absorption performance. Under in-plane quasi-static compression, it is found that higher-order HTC exhibits multi-stage deformation modes due to the non-uniform distribution of substructures, resulting in exceptional compression strength and energy absorption performance, which are increased by 30% and 200% than compared to lower order structures, respectively. Based on the deformation modes, an analytical model is derived to predict the compressive strength, which is validated by numerical and experimental results. In terms of specific energy absorption, the higher-order HTC also demonstrates enhanced performance compared to other honeycomb composite of TPU materials. Under high velocity dynamic impact, it is observed that although the multi-stage deformation mode cannot be activated for higher-order HTC, the fluctuations in dynamic response curve can still be reduced as a result of layer-by-layer collapse of substructures. Moreover, an analytical model using rigid-perfectly plastic-lock (R-PP-L) is employed to explain the dynamic behavior and predict the plateau stress under various impact velocities, which is verified by numerical simulations with good agreement. This study elucidates the effect of structural hierarchy on the mechanical properties and energy absorption performance, and the results are of potential engineering use in the design of lightweight impact-protective composites.

Successive jamming transitions in traffic flow on directed Sierpinski gasket

We study the macroscopic traffic flow and jamming transitions on directed Sierpinski gasket fractal which is closely connected to complex network. We propose a dynamic model for macroscopic traffic flow on directed Sierpinski gasket. The macroscopic dynamic equations of vehicular densities are derived on the directed network. It is shown where and how traffic jam occurs on directed networks. It is found that multiple transitions occur successively with increasing vehicular density. The successive transition points are derived numerically and analytically.

ZAGREB ECCENTRICITY INDEX OF LEVEL-n SIERPINSKI GASKET

In this paper, we investigate the Zagreb eccentricity index of the level-[Formula: see text] Sierpinski gasket [Formula: see text]. Based on the self-similarity and finite pattern, we compute the Zagreb eccentricity index of [Formula: see text] which is the integral of square of eccentricity in terms of the self-similar singular measure.

Deployment of Fractals through CNC Devices

The concept of fractal propagation, through the deployment of computer numerical controlled devices, has been discussed in this paper. For experimentation purposes, four of the most well known fractal patterns namely, Sierpinski gasket, Barnsley fern, Fractal tree and Koch snowflake curve have been chosen. This work shows the practical reproduction of before mentioned fractals on a physical media by deploying a 3-axis CNC plotter, as a part of the hardware, after proper python implementation as a part of the software. The mathematics and algorithm used in the production of the fractals have been discussed in detail and the method of deployment involves vectorisation of the fractals into a set of coordinates which are then passed on to the 3 axis CNC plotter plotter for production of the final output on a larger drawing surface. This paper aims at combining the individual concepts of fractal propagation and robotics in order to produce tangible fractal illustrations.