Sierpinski Gasket Research Articles

Some Insights into the Sierpiński Triangle Paradox

We realize that a Sierpiński arrowhead curve (SAC) fills a Sierpiński gasket (SG) in the same manner as a Peano curve fills a square. Namely, in the limit of an infinite number of iterations, the fractal SAC remains self-avoiding, such that SAC⊂SG. Therefore, SAC differs from SG in the same sense as the self-avoiding Peano curve PC⊂0,12 differs from the square. In particular, the SG has three-line segments constituting a regular triangle as its border, whereas the border of SAC has the structure of a totally disconnected fat Cantor set. Thus, in contrast to the SG, which has loops at all scales, the SAC is loopless. Consequently, although both patterns have the same similarity dimension D=ln⁡3/ln⁡2, their connectivity dimensions are different. Specifically, the connectivity dimension of the self-avoiding SAC is equal to its topological dimension dlSAC=d=1, whereas the connectivity dimension of the SG is equal to its similarity dimension, that is, dlSG=D. Therefore, the dynamic properties of SG and SAC are also different. Some other noteworthy features of the Sierpiński triangle are also highlighted.

Oscillation and Dimensions of Harmonic Functions on the Sierpinski Gaskets

Oscillation and Dimensions of Harmonic Functions on the Sierpinski Gaskets

Enumeration of multivariate independence polynomial for iterations of Sierpinski triangle graph

Enumeration of multivariate independence polynomial for iterations of Sierpinski triangle graph

Presenting the Sierpinski Gasket in Various Categories of Metric Spaces

This paper studies presentations of the Sierpinski gasket as a final coalgebra for a functor on three categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses short (non-expanding) maps, the second uses continuous maps, and the third uses Lipschitz maps. The functor in all cases is very similar to what we find in the standard presentation of the gasket as an attractor. It was previously known that the Sierpinski gasket is bilipschitz equivalent (though not isomorhpic) to the final coalgebra of this functor in the category with short maps, and that final coalgebra is obtained by taking the completion of the initial algebra. In this paper, we prove that the Sierpiniski gasket itself is the final coalgebra in the category with continuous maps, though it does not occur as the completion of the initial algebra. In the Lipschitz setting, we show that the final coalgebra for this functor does not exist.

Box dimension of harmonic functions on higher dimensional Sierpinski gasket and Sierpinski gasket with bilateral energy

This paper concentrates on the bounds of the box-counting dimension of the graph of various fractal functions on the (N-1) dimensional Sierpinski gasket for N≥3. The bounds of the box-counting dimension of the graph of harmonic functions and functions having finite energy associated with the standard self-similar energy are found. Besides that, the bounds for the graph of harmonic functions and functions having finite energy associated with the bilateral energy on the Sierpinski gasket are also examined.

Asymptotic behavior of a generalized independent sets model on the two-dimensional Sierpinski gasket

We consider a generalized independent set model on the two-dimensional Sierpinski gasket SGn. Let A be a set of symbols and each vertex of SGn should take a symbol from A. Given a subset S⊂A such that the symbols of nearest-neighbor vertices are not allowed to be the forbidden blocks F={ij:i,j∈S}. The probability of each vertex of SGn to take a symbol in S is denoted by p ∈ (0, 1), and the probability of each vertex to take a symbol not in S is given by 1 − p. When A={0,1}, S={0}, and p = 1/2, this reduces to independent set model or golden mean shift in symbolic dynamics. We investigate the asymptotic behavior of this generalized model on the two-dimensional Sierpinski gasket and obtain upper and lower bounds of the entropy per site for p ∈ (0, 1).

Design and Analysis of Microstrip Sierpinski Fractal Antenna for Wireless application

This paper describes a novel design for a microstrip fractal antenna based on the Sierpinski triangle shape. It is built on a FR4 substrate and operates in the 5.5 GHz frequency range. The proposed antenna is designed and validated using ANSYS Electronic Desktop's High Frequency Structure Simulator (HFSS). The simulated results show good performance in terms of radiation pattern, gain and input impedance. This proposed antenna can be widely used in wireless communication equipment that is progressing towards miniaturization and high frequency. The antenna designed gives a return loss of -29 dB at 5.53GHz. It has a gain of 0.85 dB and directivity of 1.8dB.

Many-body localization on finite generation fractal lattices

We study many-body localization in a hardcore boson model in the presence of random disorder on finite generation fractal lattices with different Hausdorff dimensions and different local lattice structures. In particular, we consider the Vicsek, T-shaped, Sierpinski gasket, and modified Koch-curve fractal lattices. In the single-particle case, these systems display Anderson localization for arbitrary disorder strength if they are large enough. In the many-body case, the systems available to exact diagonalization exhibit a transition between a delocalized and localized regime, visible in the spectral and entanglement properties of these systems. The position of this transition depends on the Hausdorff dimension of the given fractal, as well as on its local structure.

Fractional Domination Number of the Fractal Graphs

Background/Objectives: A fractal is a collection of clearly defined structures that are uneven or irregular and can be divided into smaller pieces. Each of these smaller pieces has a relationship to the overall structure at random ranges that are analogous or identical. Fractal graphs are a highly effective way to construct well-defined patterns. The fractional domination number in a graph refers to the weight of a minimum fractional dominating function. The study is to find the bounds of the fractional domination and their related parameters in the fractal graphs. Methods: The sharp bounds and the constant ratio of fractional domination and their related parameters were calculated using an iterative process in all iterations of a self-similarity fractal graph. Findings: The study established the relation between the fractional domination number and other relevant parameters as fractional independence domination numbers for each iteration in the fractal graphs and explained how the concept of fractional domination works on fractal graphs such as the Koch snowflake and the Sierpinski triangle. Novelty: Previous researchers have studied bounds on maximum matching and dominating sets in fractal graphs such as the Koch snowflake and the Sierpinski triangle. The present study enhances the idea of finding the bounds of fractional domination parameters in a fractal graph. The construction of new beautiful ornaments, locating damaged cells for drug injection in medical science, placing stones in jewels, plumping work, and designing fractal antennas all can be made possible by this work in self-similarity fractal graphs. Keywords: Fractals, Fractional Dominating Functions, Fractional Domination Number, Fractional Independence Domination Number, Fractional Domination Chain

A new notion of subharmonicity on locally smoothing spaces, and a conjecture by Braverman, Milatovic, Shubin

Given a strongly local Dirichlet space and λ⩾0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\geqslant 0$$\\end{document}, we introduce a new notion of λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-subharmonicity for Lloc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1_\ extrm{loc}$$\\end{document}-functions, which we call localλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-shift defectivity, and which turns out to be equivalent to distributional λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-subharmonicity in the Riemannian case. We study the regularity of these functions on a new class of strongly local Dirichlet, so called locally smoothing spaces, which includes Riemannian manifolds (without any curvature assumptions), finite dimensional RCD spaces, Carnot groups, and Sierpinski gaskets. As a byproduct of this regularity theory, we obtain in this general framework a proof of a conjecture by Braverman, Milatovic, Shubin on the positivity of distributional Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}-solutions of Δf⩽f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta f\\leqslant f$$\\end{document} for complete Riemannian manifolds.

A Narrative Approach to Foster the Construction of Recursive Thinking in High School Students

In this paper, we will explain the development of a mathematical activity involving narrative and short stories in particular, with the aim of investigating whether it is possible to use the narrative approach to promote the construction of recursive thinking in high school students from a four-year scientific high school (Grades 11 and 12). We present qualitative research based on the networking of two theoretical frameworks used to analyze students’ protocols and the issues surfacing during class discussion: Abstraction in Context (AiC) and Documenting Collective Activity (DCA). In our research, the students, divided into small groups, dealt with a highly immersive “story problem” with the Sierpinski Triangle as its central element. The task was designed to ensure consistency with the story and involved the construction, with GeoGebra 6.0 software, of a fractal city, Fractlandia, with squares and sinkholes. The preliminary results show that the story proposed functioned as a motivation to solve the problem, and the last questions of the task proved the most engaging for the students, mainly because of the connection with the story, and also because they involved some reflection about the behavior to the infinity of the perimeter and the area of the Sierpinski Triangle.

Fractal Inspired Hybrid Microstrip Patch Antenna for Surveillance Drone Applications

Drone based surveillance and communication systems are playing a vital role in modern days for both military and civilian applications. Antennas with low profile, smaller size, and light weight are essential for the realization of these systems. A novel fractal-inspired hybrid microstrip patch antenna realized with Sierpinski gasket fractal slots cut in the radiating patch along with corresponding ground structure simultaneously defected with Sierpinski carpet fractal slots is reported for the first time. The antenna resonates at 2.4 GHz with a measured return loss of 17 dB, gain of 7 dB and measured 3 dB beamwidth is of the order of 70 deg. The current approach leads to the size reduction of the antenna to the extent of 55% in comparison to standard radiating patch antenna resonating at the same frequency. The fractal-inspired hybrid microstrip patch antenna is a promising candidate for drone-based surveillance and communication applications owing to its miniaturization and good directional radiation properties.

Recent Developments in Iterative Algorithms for Digital Metrics

This paper aims to provide a comprehensive analysis of the advancements made in understanding Iterative Fixed-Point Schemes, which builds upon the concept of digital contraction mappings. Additionally, we introduce the notion of an Iterative Fixed-Point Schemes in digital metric spaces. In this study, we extend the idea of Iteration process Mann, Ishikawa, Agarwal, and Thakur based on the ϝ-Stable Iterative Scheme in digital metric space. We also design some fractal images, which frame the compression of Fixed-Point Iterative Schemes and contractive mappings. Furthermore, we present a concrete example that exemplifies the motivation behind our investigations. Moreover, we provide an application of the proposed Fractal image and Sierpinski triangle that compress the works by storing images as a collection of digital contractions, which addresses the issue of storing images with less storage memory in this paper.

Average trapping time on horizontally divided 3-dimensional 3-level Sierpinski gasket network

An important characteristic of random wandering is the average trapping time, which is a hot issue in current research. The average trapping time is an important measure of the transmission efficiency of random wandering in a network. In this paper, we construct a 3-dimensional 3-level Sierpinski gasket network divided horizontally by the horizontal division plane P s , that is, the division coefficients s. We study the capture problem on the network and obtain an analytical expression for the average trapping time (ATT). Then, by adjusting the number of iterations and the values of the division coefficients, we obtained the relationship between ATT and them. As can be seen from our numerical simulation plots, ATT is affected by s. The larger s is, the more the self-similar structure of the three-dimensional residual network gradually transforms towards the structure of the two-dimensional complete Sierpinski gasket network. Meanwhile, the shorter ATT is, that is, the more efficient the transmission on the network.

Characterization of the effect of fractal micro-protrusion and heat on the wettability behavior on chips in medical MEMS

Interfacial wettability, a fundamental property of solid surfaces in the medical field, is known to be affected by surface microstructure and temperature. This research aims to examine the influence of temperature on wettability by creating two distinct types of fractal single-stage microstructures on medical silicon wafers, utilizing the combined features of single-stage microstructure and fractal micro-nano structure. The experimental findings indicate that the area fraction and roughness of the fractal microstructure exhibit variations based on the specific fractal dimension employed. The hydrophobicity of the Sierpinski triangle fractal microstructure surpasses that of other tested microstructures under the same machining accuracy, which can increase the hydrophobicity of the chip surface by 47.9% on average. Furthermore, the hydrophobicity of the Sierpinski triangle fractal microstructure is found to be 13.1% higher than that of the Sierpinski carpet fractal microstructure, despite the temperature stability of the latter is 35% higher. When compared to a uniform micropillar structure with identical parameters, the Sierpinski carpet fractal microstructure demonstrates superior hydrophobicity, while the Sierpinski triangle fractal microstructure exhibits lower hydrophobicity. These findings suggest that the incorporation of fractal microstructures significantly influences the wettability of medical chips under different temperatures, which holds promise for its application in medical MEMS.

Total effects with constrained features

Recent studies have emphasized the connection between machine learning feature importance measures and total order sensitivity indices (total effects, henceforth). Feature correlations and the need to avoid unrestricted permutations make the estimation of these indices challenging. Additionally, there is no established theory or approach for non-Cartesian domains. We propose four alternative strategies for computing total effects that account for both dependent and constrained features. Our first approach involves a generalized winding stairs design combined with the Knothe-Rosenblatt transformation. This approach, while applicable to a wide family of input dependencies, becomes impractical when inputs are physically constrained. Our second approach is a U-statistic that combines the Jansen estimator with a weighting factor. The U-statistic framework allows the derivation of a central limit theorem for this estimator. However, this design is computationally intensive. Then, our third approach uses derangements to significantly reduce computational burden. We prove consistency and central limit theorems for these estimators as well. Our fourth approach is based on a nearest-neighbour intuition and it further reduces computational burden. We test these estimators through a series of increasingly complex computational experiments with features constrained on compact and connected domains (circle, simplex), non-compact and non-connected domains (Sierpinski gaskets), we provide comparisons with machine learning approaches and conclude with an application to a realistic simulator.

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.

Probing fractal spatiotemporal inhomogeneity in a quantum walk

We investigate the transport and entanglement properties exhibited by quantum walks with coin operators concatenated in a space-time fractal structure. Inspired by recent developments in photonics, we choose the paradigmatic Sierpinski gasket. The 0-1 pattern of the fractal is mapped into an alternation of the generalized Hadamard-Fourier operators. This two-state coin-operator approach overcomes the intricacies caused by the utilization of high-dimensional coin operators required in prior studies of discrete-time quantum walks on fractals. In fulfilling the blank space on the analysis of the impact of inhomogeneity in quantum walk properties, specifically, fractal deterministic inhomogeneity, our results show a robust effect of entanglement enhancement as well as an interesting road to superdiffusive spreading with a tunable scaling exponent attaining robust superdiffusion, subballistic though. Explicitly, with this fractal approach it is possible to obtain an increase in quantum entanglement with reduced impact on standard quantum walk theoretical spreading. Alongside those features, we analyze further properties such as the degree of interference and visibility. The present model corresponds to an application of fractals in an experimentally viable setting, namely, the building block for the construction of photonic patterned structures. Published by the American Physical Society 2024

A uniform trace theorem for Dirichlet forms on Sierpinski fractals

We establish trace theorems for the self-similar Dirichlet forms on the Sierpinski gasket and the Sierpinski carpet to their subsets generated by cutting with a straight line. For the Sierpinski gasket, the straight line can be in any direction. For the Sierpinski carpet, we require the straight line parallel to an edge of the carpet. The trace forms are expressed in term of values of functions along the cut, in a uniform way independent of the choices of the line.

OPTICAL FEEDBACK. I STATIC EFFECTS.

Feedback is a common phenomenon. This paper gives examples of feedback in computer science (iterations). The primary aim of this paper is to observe the effects of optical feedback in a monitor-camera system. The static effects of optical feedback are presented in this paper. The problem of symmetry of the obtained images is discussed. It is explained how a multiple three-lens copying machine generates the Sierpinski triangle, which is a basic and very well-known fractal. In the next paper ( Part II ) the dynamic effects of optical feedback will be discussed.