Sierpinski Gasket Research Articles

Behavioral fractal method associated with GPS tracking to spatial activity sequences of grazing cattle

Global positioning and geographic information systems are essential for studying foraging animal behavior. This study aims to implement a fractal self-similarity and chaos game computational efficient methodology to determine the behavior-associated fractal using GPS data of activity sequences in spatial ranges of livestock movement trajectories in interaction with habitat factors. Six cows were randomly selected with an average weight of 480 kg, maintained under the same conditions, and a GPS-equipped collar was installed, programmed at intervals of 1 min and an average of 9 h daylight. Roughly 192810 registries and an average of 32135 signals per cow from trajectory tracking in spatial activity sequencing were used as a variable of interest in the fractal characterization methodology. Spatial patterns were evaluated using the Morán’s spatial autocorrelation indices, cluster, and non-parametric statistics, evaluating deterministic spatial patterns of preferential activities associated to spatial ranges of less than 7.1 m (resting 42 %, grazing 38 %). GPS information was refined through spatial ranges and changes in activities under resting, eating, traveling, and complementary schemes associated to the fractal displacement behavior of grazing cattle. This information was processed and mapped using fractal self-similarity rules in the Sierpinski triangle to determine the typical fractal of spatial activities per animal in the habitat. The particular fractal record of each bovine as a function of trajectory sequences was mapped for binary image matrices, registering a good classification (83 %) of the animals by breed and climatological cycle, using information from the sequencing of spatial activities associated to the preferred behavior in the habitat.

Code properties of the holographic Sierpinski triangle

We study the holographic quantum error correcting code properties of a Sierpinski triangle-shaped boundary subregion in ${\mathrm{AdS}}_{4}/{\mathrm{CFT}}_{3}$. Due to existing no-go theorems in topological quantum error correction regarding fractal noise, this gives holographic codes a specific advantage over topological codes. We then further argue that a boundary subregion in the shape of the Sierpinski gasket in ${\mathrm{AdS}}_{5}/{\mathrm{CFT}}_{4}$ does not possess these holographic quantum error correction properties.

Hitting time for random walks on the Sierpinski network and the half Sierpinski network

We consider the unbiased random walk on the Sierpinski network (Sn◦N) and the half Sierpinski network (HSn◦N), where n is the generation. Different from the existing works on the Sierpinski gasket, Sn◦N is generated by the nested method and HSn◦N is half of Sn◦N based on the vertical cutting of the symmetry axis. We study the hitting time on Sn◦N and HSn◦N. According to the complete symmetry and structural properties of Sn◦N, we derive the exact expressions of the hitting time on the nth generation of Sn◦N and HSn◦N. The curves of the hitting time for the two networks are almost consistent when n is large enough. The result indicates that the diffusion efficiency of HSn◦N has not changed greatly compared with Sn◦N at a large scale.

Non‐homogeneous p‐Laplacian equations on the Sierpinski gasket

Let be the Sierpiński gasket in and denote the boundary of . In this paper, we study the following non‐homogeneous ‐Laplacian equation where are real numbers such that and the function is suitably chosen. The existence of at least two nontrivial weak solutions to the above non‐homogeneous equation on the Sierpiński gasket will be established.

Generating Sierpinski gasket from matrix calculus in Dempster–Shafer theory

Graphics with fractal features are usually generated using the Iterated Function System (IFS). IFS can generate the entire family of Sierpinski gaskets by performing different operations on the attractors. As the most classical graphic, Sierpinski gasket can also be generated using mod(n,2). Dempster–Shafer Theory (DST), as a mathematical theory about evidence, models information on the all possible combination states (power set), which relates to 2n. In this paper, we explore the relationship between the Sierpinski gasket and matrix calculus in DST, which is the first time to connect fractal theory and DST from the perspective of geometry. In addition, based on the generation process of the matrices, we propose a method to generate the Sierpinski Gasket using the Kronecker product.

GPF Design Based on Sierpinski Triangle

In order to adapt to strict emission regulations and reduce particulate emissions from gasoline engines, continuous research on GPF is needed to improve particulate capture efficiency. The structure and working principle of gasoline engine particle catchers are studied. A new type of GPF is designed based on the Sierpinski triangle structure. The GPF shell is designed according to the structure of the filter body, and the flow guide plate is designed to make the gas flow rate uniform. The model and accessories of the filter body are modeled in Solid Works. Based on the repeated characteristics of the filter body structure, a unit body is extracted to simplify the analysis. The simulation was performed using the filter unit. The flow rate of exhaust gas between the new filter structure and the traditional filter body was compared in addition to the pressure of the filter body and the trapping effect of small molecular particles in the exhaust gas. Through analysis and comparison, it can be concluded that the new GPF filter structure has better trapping efficiency than the traditional filter structure.

Design of a Compact Reconfigurable Sierpinski Gasket Fractal Antenna for 5G and Satellite Communications

Reconfigurability is one of the key features of 5G/6G radiofrequency systems. The possibility to adapt the working frequency depending on the application or the status of the nodes is important, especially in Smart Radio Environments. In this regard, the design of antennas operating at different frequency bands is challenging since strict requirements on the footprints and the radiation properties are introduced. Among other geometries, fractals present unique features in terms of small electrical lengths: they have been used for years by researchers to shrink antenna dimensions in a very effective way. An interesting perspective of these antennas is the possibility to switch the working frequency among several bands connecting/disconnecting the fractal cells. The aim of this paper is the design and the optimization of an electrically reconfigurable Sierpinski gasket fractal antenna. Four different states are obtained: in the first the device works at 27 GHz (mmWave), in the second at 16 GHz, in the third at 11 GHz (Ku-band), and in the fourth at 3.5 GHz (Sub-6GHz). The antenna operates efficiently in all the bands of interest.

AVERAGE TRAPPING TIME ON THE 3-DIMENSIONAL 3-LEVEL SIERPINSKI GASKET NETWORK WITH A SET OF TRAP NODES

As a basic dynamic feature on complex networks, the property of random walk has received a lot of attention in recent years. In this paper, we first studied the analytical expression of the mean global first passage time (MGFPT) on the 3-dimensional 3-level Sierpinski gasket network. Based on the self-similar structure of the network, the correlation between the MGFPT and the average trapping time (ATT) is found, and then the analytical expression of the ATT is obtained. Finally, by establishing a joint network model, we further give the standard process of solving the analytical expression of the ATT when there is a set of trap nodes in the network. By illustrating examples and numerical simulations, it can be proved that when the trap node sets are different, the ATT will be quite different, but the super-linear relationship with the number of iterations will not be changed.

On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set

We investigate Benford’s law in relation to fractal geometry. Basic fractals, such as the Cantor set and Sierpinski triangle are obtained as the limit of iterative sets, and the unique measures of their components follow a geometric distribution, which is Benford in most bases. Building on this intuition, we aim to study this distribution in more complicated fractals. We examine the Laurent coefficients of a Riemann mapping and the Taylor coefficients of its reciprocal function from the exterior of the Mandelbrot set to the complement of the unit disk. These coefficients are 2-adic rational numbers, and through statistical testing, we demonstrate that the numerators and denominators are a good fit for Benford’s law. We offer additional conjectures and observations about these coefficients. In particular, we highlight certain arithmetic subsequences related to the coefficients’ denominators, provide an estimate for their slope, and describe efficient methods to compute them.

Métodos de diferencias finitas y elementos finitos para ecuaciones diferenciales parciales sobre dominios fractales.

In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms of the equation derived using standard length or área measure on a discrete approximation of the fractal set. We then introduce a numerical procedure to normalize the obtained diffusions, that is, a way to compute the renormalization constant needed in the definitions of the actual partial differential equation on the fractal set. A particular case that is studied in detail is the solution of the Dirichlet problem in the Sierpinski triangle. Other examples are also presented including a non-planar Hata tree.

Thank you to our referees April 8, 2021–May 31, 2022

I’ve always been fascinated by fractals, especially the Sierpinski gasket, shown at right, and in a beautiful pyramidal arrangement in the background of this spread. To make such a gasket, all you do is remove the interior of the middle equilateral triangle (the upside down one) over and over again, at successively smaller and smaller scales. What results is something that is more than one dimensional, but less than two dimensional—in fact, it can be shown to have dimension D, where D is given by the relation 2D=3, or D ≈ 1.585…. In this issue of TPT, we feature on page 410 a paper by Charles Creffield, “Fractals on a Benchtop: Observing Fractal Dimension in a Resistor Network,” where he details how his students set up resistors in a similar network and mimic fractal resistance effects … doubly fascinating! I wanted to share this imagery with you in appreciation of the hundreds of generous, smart, and gracious individuals who serve as TPT referees. When I look over this list of names, I see many of the people I admire most in the world, and it makes me optimistic in a way that few things can these days. If you see someone you know in this list, please find a way to express your thanks for all they do for TPT and for physics teachers and students everywhere. When I imagine all the connections between the people on this list, it calls to my mind all of the connections between the dark regions in the figure at right—a Sierpinski gasket of physicists, held together by their common ideas. What could be more fun to think about!? Thank you, again, dear referees; you are the glue that holds this community together, in my view.

Topological Properties, Spectra Analysis, and Consensus Problems for a Class of Network Models Based on m-Fission Operation

The network model, especially the network model constructed by network operation, is an important tool to study complex networks. Many topological and dynamic properties of complex networks can be studied in this way. In this article, the m -fission operation is constructed based on the phenomenon of node splitting in the network, which is quite common in complex networks. Many network models, including dual Sierpinski gaskets, are built based on this operation, but it has never been systematically studied. Then, the topological and dynamic properties of the m -fission operation and the corresponding iterative fission network model are studied, and the influence of the operation on the network structure is revealed. Among them, the topological properties of the network include diameter, degree distribution, clustering coefficient, average distance, and modularity. By studying these properties, it can be concluded that the iterative fission network is a fractal homogeneous network with high clustering and high community characteristics. Since the dynamic properties are closely related to the spectrum of the Laplacian matrix corresponding to the network, the iterative relation of the spectrum in operation is studied, and the complete solution of the spectrum of the iterative fission network is obtained. Based on the above results, we calculate the analytical expressions of the characteristic quantities related to the dynamic properties on the network, including the Kirchhoff index and the average of hitting times. Finally, due to the close connection between the network model and the system, we further analyzed the consensus problems on the system corresponding to the network, including convergence rate, delay robustness, first-order noise coherence, and second-order noise coherence.

Convergence of Energy Forms on Sierpinski Gaskets with Added Rotated Triangle

We study the convergence of resistance metrics and resistance forms on a converging sequence of spaces. As an application, we study the existence and uniqueness of self-similar Dirichlet forms on Sierpinski gaskets with added rotated triangle. The fractals depend on a parameter in a continuous way. When the parameter is irrational, the fractal is not post critically finite (p.c.f.), and there are infinitely many ways that two cells intersect. In this case, we define the Dirichlet form as a limit in some Γ-convergence sense of the Dirichlet forms on p.c.f. fractals that approximate it.

Capacitance for fractal-like disordered dielectric slab

In this paper, we model a heterogeneous dielectric medium exhibiting fractal geometry or disordered random structures by applying non-integer dimensions to determine its capacitance between two parallel plates. The capacitance depends on the fractional dimensions of the fractal or disordered dielectric slab, which may be obtained from the theoretical fractal dimension or box-counting method. The findings are verified by CST Studio Suite (Electromagnetic field simulation software), experimental measurements, and the equivalent capacitance method. Five common types of fractals (Cantor bars/plates, Sierpinski carpet, Sierpinski triangle, Haferman carpet, and Menger sponge) and random structures are tested with good agreement. There is also an effective gain of capacitance in using less amount of dielectric materials, which may be useful in material-savings of dielectrics. This research shows a useful tool in modeling the capacitance of heterogeneous materials, where fractals and disordered structures may be commonly encountered in organic materials and any dielectrics where precision and fabrication are not perfect.

Application of Selected Fractal Geometry Resonators in Microstrip Strain Sensors

Until now, strain gauges, fiber optic sensors, magnetic sensors, and piezoelectric sensors have been used for strain measurements. In recent years, the idea of using microstrip antennas in strain measurement applications has emerged. One of the key requirements for deformation sensors is the ability to mount them on small-size elements (therefore the size of these sensors is crucial). The possible way to miniaturize microstrip sensors is the application of appropriate patch geometry, e.g. fractal geometry. In this article application of various fractal patch geometries in microstrip strain sensors were investigated. All sensors were designed for the same operating frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${f}_{\textbf {r}} =2.725$ </tex-math></inline-formula> GHz and on the same laminate. This approach showed how the influence of a specific geometry affects the sensitivity and the transducer size. The Sierpinski carpet, the Sierpinski triangle, the Sierpinski curve, and the combination of the Koch curve and the Sierpinski carpet were used to obtain the fractal geometry and miniaturize the sensor dimensions. They have been compared with the widely used rectangular patch. Numerical and experimental analyzes for the proposed sensors were carried out. The best solution to this problem was to use a combination of the Koch curve and the Sierpinski carpet. This fractal geometry, compared to a rectangular patch, enables the reduction of the patch size by 70% while reducing the sensitivity by 26%.

Sound-scattering properties of Sierpinski triangle fractal structures in the near field

This paper proposes a new design of an acoustic diffuser based on the construction rules of the Sierpinski triangle in order to broaden the effective diffusion frequency range. The diffuser is made of triangular blocks of different sizes attached to a plane surface. The effects of the number of fractal iterations, the height of triangular blocks, and arrangements of the blocks on the normal-incidence diffusion coefficients in the near field are examined through numerical simulations based on the boundary element method (BEM) in the frequency range of 100 Hz – 5 kHz. Furthermore, measurement results will be presented to validate the diffusion performance presented by the numerical simulations. The diffusion performance of a conventional quadratic residue diffuser (QRD) is compared to confirm the advantage of the designed diffuser for broadening the effective frequency range. It shows that the fractal patterns with various sizes of blocks improve diffusion performance compared to the conventional QRD of the same size, especially in the mid-low frequency range below 1 kHz.

A Proposal, Design, and Analysis of Planar Feed Gasket Antennas with Modified Substrate for Multiband Vehicular Applications

The Sierpinski Gasket fractal antennas can provide the best generic multiband behaviour among the several planar feed monopole antennas available for use in automotive applications. The Gasket geometries used in antennas can be classical, scale perturbed, or both scale and flare perturbed. When used in the planar feed configuration, the classical and perturbed Gasket fractal antennas suffer from considerable reflections at one or more multiband frequencies, rendering these frequencies unsuitable for usage. Therefore, a design modification in these antennas is essential. A novel design modification has been proposed in the Sierpinski Gasket fractal antennas in the present research work by utilizing metamaterial engineered substrates. It has been found that the metamaterials not only reduce reflections but these materials also lower Gasket antenna frequencies, thereby, facilitating reduction in the antenna size. The findings have been validated through the experimental investigations. The comprehensive design analysis presented in the present paper shall prove highly beneficial to the antenna designers in rapidly customizing the vehicular antennas.

The study of the b-choromatic number of some classes of fractal graphs

In graph coloring, labels are assigned to graph elements according to certain constraints. Colors are a special case of graph labeling as well as in practical applications, graph coloring also poses some theoretical challenges. A topic related to graph coloring will be discussed in this study, i.e., b-chromatic number. In proper coloring, edges, vertices, or both of them are colored so that they are distinct from one another. A b-coloring of m colors of a graph G is similar to proper coloring in which at least one vertex from each color class is connected to (m-1) other colors. The b-chromatic number of a graph G is the greatest positive number k such that G admits a b-coloring with k colors and is represented by ϕ(G). Fractals are geometric objects that are self-similar at multiple scales and their geometric measurements are different from fractal measurements. In this paper, we will evaluate the b-chromatic number of Fractal type graphs, i.e., Sierpinski network S(n; Kk) (where Kk is a complete graph of order k) and Sierpinski gasket network S(n). Firstly, we will compute the b-chromatic number of S(n; K3), S(n; K4) and S(n; K5) for n ≥ 2. After that, we will generalize the result for the Sierpinski network of complete graph Kk. In addition, we will also determine the b-choromatic number of Sierpinski gasket graph S(n). As an application, we will also determine the b-chromatic number of Sierpinski graph of house graph.

An IFS for the Stretched Level-n Sierpinski Gasket

Broadly, fractals are sets that exhibit a repeating pattern at multiple scales. One important fractal set is the Sierpinski Gasket (SG) which is made up of nested equilateral triangles. A variation of the classic Sierpinski gasket is to create n-levels with the equilateral triangles. Another variation is to stretch the points of intersection for the triangles in SG into line segments of length 0 &lt; α &lt; 1/3. When one combines these variations, one arrives at the stretched level-n Sierpinski gaskets (SSGn) which are the focus of this work. We give an introduction to iterated function systems (IFS) and determine an IFS which generates SSG3. We then describe how one can acquire the IFS for SSGn in general, and conclude with a theorem which determines the Hausdorff dimension for SSGn.

Stochastic modelling of fractal diffusion and dimension estimation

The revision of classical methods for spectral and walk dimension estimates is the main aim of the paper. Being focused on the unbiased estimation of the walk and spectral dimensions, we aim to construct the estimates with the minimal mean square error. Accompanied simulation experiments are performed on finite substrates, spacial structures serving as a good model of both continuum and fractal sets. We compare the classical approach based on the log–log transform of asymptotic models of returning probabilities and the second moments, and we also develop a weighted approach to improve the statistical properties of dimension estimates. The other discussed aspect is whether to simulate diffusion using the classical graph diffusion model with zero probability of staying in the same vertex or to prefer the physically motivated model of diffusion over edges with the optimal value of jump probability. Finally, we present the results of simulation experiments on two-dimensional finite substrates which approximate the continuum and selected Sierpinski gaskets and carpets. The paper also summarises general suggestions based on the obtained results from the simulation experiments.