Koch Fractal Research Articles

The Hausdorff Dimension and Capillary Imbibition

The time scaling exponent for the analytical expression of capillary rise ℓ∼tδ for several theoretical fractal curves is derived. It is established that the actual distance of fluid travel in self-avoiding fractals at the first stage of imbibition is in the Washburn regime, whereas at the second stage it is associated with the Hausdorff dimension dH. Mapping is converted from the Euclidean metric into the geodesic metric for linear fractals F governed by the geodesic dimension dg=dH/dℓ, where dℓ is the chemical dimension of F. The imbibition measured by the chemical distance ℓg is introduced. Approximate spatiotemporal maps of capillary rise activity are obtained. The standard differential equations proposed for the von Koch fractals are solved. Illustrative examples to discuss some physical implications are presented.

Design and Geometric Transformations of Koch Curve Monopole Antennas

In this paper, a modified Koch curve monopole antenna is proposed for the dual-band Wireless Local Area Network (WLAN) and its performance has been compared with the conventional printed Koch-curve monopole antenna. Both the antennas have been contrived, and their simulated return loss and radiation pattern results have been validated with the measurements. Antenna #1 is designed on the geometrical basis of a conventional Koch-curve monopole antenna. Moreover, the shape of antenna # 1 is bent with a two-step rotation process to yield antenna #2. There are two advantages to this geometrical modification: Firstly, the unique rotation in antenna #2 has further increased the mutual coupling between the segments of the Koch-curve geometry, and thus improved the impedance matching. Secondly, due to its geometric configuration that enhanced its ground plane size, the proposed antenna #2 has been able to demonstrate consistent simulation and measurement results. The compact ground plane in antenna #1 has been identified as the primary reason for the disagreement between the simulated and measured return loss results. The presented antennas have also been studied for the variation of indentation angles: θ=30 and θ=60.

RETRACTED ARTICLE: Internet of Thing based Koch Fractal Curve Fractal Antennas for Wireless Applications

We, the Editor, Institution and Publisher of the journal IETE Journal of Research, have retracted the following article, which is part of the Special Issue titled “Federated Learning for Blockchain Systems and Industrial Internet of Things”: Kusum Yadav, Anurag Jain, Nada Mohamed Osman Sid Ahmed, Sawsan Ali Saad Hamad, Gaurav Dhiman & Shoayee Dlaim Alotaibi (2022) Internet of Thing based Koch Fractal Curve Fractal Antennas for Wireless Applications, IETE Journal of Research, doi:10.1080/03772063.2022.2058631 Following publication, concerns were raised about the peer review and decision-making processes for this special issue. After an investigation by the Taylor & Francis Publishing Ethics & Integrity team, in full cooperation with the Editor-in-Chief and the Institution, it was confirmed that the articles included in this special issue were not reviewed appropriately, in line with the Journal’s peer review standards and policy. As the stringency of the peer review process is core to the integrity of the publication process, the Editor and Publisher have decided to retract all of the articles within the above-named Special Issue. The journal has not confirmed if the authors were aware of this compromised peer review process. The journal is committed to correcting the scientific record and will fully cooperate with any institutional investigations into this matter. The authors have been informed of this decision. We have been informed in our decision-making by our editorial policies and the COPE guidelines. The retracted articles will remain online to maintain the scholarly record, but they will be digitally watermarked on each page as “Retracted”.

Batik Jlamprang with Koch Snowflake and Koch Anti Snowflake Fractal Geometry using Desmos

Batik Jlamprang is a cultural heritage from Pekalongan. This batik has a round shape and floral ornaments. The motif of batik Jlamprang is similar to the Koch snowflake. Mathematically, batik Jlamprang is one of the shapes of fractal geometry. There are many known shapes of fractals, some of which are Koch Snowflake and Koch Anti-Snowflake. The difference between Koch Snowflake and Koch Anti-Snowflake lies in the generating process. Koch Anti-Snowflake is the opponent of Koch Snowflake. The main step of the process is generated by compiling Koch Snowflake and Koch Anti-Snowflake function formulas, followed by iteration. The making of the batik motif is initially carried out traditionally, which has disadvantages in terms of time and cost. However, nowadays, the motif of batik Jlamprang can be made mathematically with the help of Desmos software. This will definitely shorten the time and reduce production costs. Desmos software was chosen because it has several advantages, including being easy to operate via mobile phone or computer. This paper examines the function formula, iteration, and application of Koch Snowflake and Koch Anti-Snowflake fractal geometry in designing batik Jlamprang assisted by Desmos. The method used was library research by collecting several relevant sources. The generating process produced a function formula of Pn (perimeter) and An or Sn (area) which is useful for designing the motif of batik Jlamprang. The visualization process was carried out on Desmos, followed by geometry transformation and cloning that were able to produce the motif of batik Jlamprang as desired.

A novel planar differential excitation eddy current probe based on the fractal Koch curve

Planar eddy current (EC) probe is one of the most important EC probe for detecting the defects in conductive parts. A novel planar differential excitation EC probe are proposed and its excitation coil is obtained by the topological transformation combining the fractal Koch curve. To explore the right combination of the excitation coil and pickup coil and compare the performance of the probes, ten different configurations of the EC probe are studied by the simulations and experiment. The results show that the best pickup coil of each excitation is with the same shape and size of the excitation; The Koch excitation and Koch pickup probe have higher sensitivity for 3mm and 5mm length defects in the special direction than other configurations.

Fractal analysis of anatomical structures linear contours: modified Caliper method vs Box counting method

Fractal analysis estimates the metric dimension and complexity of the spatial configuration of different anatomical structures. This allows the use of this mathematical method for morphometry in morphology and clinical medicine. Two methods of fractal analysis are most often used for fractal analysis of linear fractal objects: the Box counting method (Grid method) and the Caliper method (Richardson’s method, Perimeter stepping method, Ruler method, Divider dimension, Compass dimension, Yard stick method). The aim of the research is a comparative analysis of two methods of fractal analysis – Box counting method and author's modification of Caliper method for fractal analysis of linear contours of anatomical structures. A fractal analysis of three linear fractals was performed: an artificial fractal – a Koch snowflake and two natural fractals – the outer contours of the pial surface of the human cerebellar vermis cortex and the cortex of the cerebral hemispheres. Fractal analysis was performed using the Box counting method and the author's modification of the Caliper method. The values of the fractal dimension of the artificial linear fractal (Koch snowflakes) obtained by the Caliper method coincide with the true value of the fractal dimension of this fractal, but the values of the fractal dimension obtained by the Box counting method do not match the true value of the fractal dimension. Therefore, fractal analysis of linear fractals using the Caliper method allows you to get more accurate results than the Box counting method. The values of the fractal dimension of artificial and natural fractals, calculated using the Box counting method, decrease with increasing image size and resolution; when using the Caliper method, fractal dimension values do not depend on these image parameters. The values of the fractal dimension of linear fractals, calculated using the Box counting method, increase with increasing width of the linear contour; the values calculated using the Caliper method do not depend on the contour line width. Thus, for the fractal analysis of linear fractals, preference should be given to the Caliper method and its modifications.

A Wideband Hybrid Fractal Ring Antenna for WLAN Applications

We propose the design of a novel fractal antenna that is both unique and performance-driven. Two important antenna design features, miniaturization and wideband operation, are combined in this work. A ring-shaped antenna is designed using the well-known fractal geometry. This hybrid geometry is a fusion of meander and Koch curve shapes. The geometrical construction of the proposed antenna is compared to the standard Koch curve geometry. It is shown that combining the meander and Koch curve shapes increases the effective electrical length. The wider bandwidth is achieved by bringing the higher modes together. The overall dimensions of proposed meander Koch curve fractal ring antenna are 45 × 25 × 1.6 mm3. The resonance frequency of the antenna is between 4.94 and 6.12 GHz (% BW = 21.83), which covers the entire 5 GHz WLAN band. The prototype has been fabricated and experimentally verified.

Acoustic scattering by impedance screens/cracks with fractal boundary: Well-posedness analysis and boundary element approximation

We study time-harmonic scattering in [Formula: see text] ([Formula: see text]) by a planar screen (a “crack” in the context of linear elasticity), assumed to be a non-empty bounded relatively open subset [Formula: see text] of the hyperplane [Formula: see text], on which impedance (Robin) boundary conditions are imposed. In contrast to previous studies, [Formula: see text] can have arbitrarily rough (possibly fractal) boundary. To obtain well-posedness for such [Formula: see text] we show how the standard impedance boundary value problem and its associated system of boundary integral equations must be supplemented with additional solution regularity conditions, which hold automatically when [Formula: see text] is smooth. We show that the associated system of boundary integral operators is compactly perturbed coercive in an appropriate function space setting, strengthening previous results. This permits the use of Mosco convergence to prove convergence of boundary element approximations on smoother “prefractal” screens to the limiting solution on a fractal screen. We present accompanying numerical results, validating our theoretical convergence results, for three-dimensional scattering by a Koch snowflake and a square snowflake.

Numerical Simulations of Magnetohydrodynamics Natural Convection and Entropy Production in a Porous Annulus Bounded by Wavy Cylinder and Koch Snowflake Loaded with Cu-Water Nanofluid.

The current paper presents a numerical study of the magnetohydrodynamics natural convection and entropy production of Cu–water nanofluid contained in a porous annulus between a heated Koch snowflake and wavy cylinder with lower temperature with respect to the Koch snowflake. The numerical algorithm is based on the Galerkin Finite Element Method. The impacts of Rayleigh number (Ra = 103, 104, 105, and 106), Hartman number (Ha = 0, 25, 50, and 100), Darcy number (Da = 10−2, 10−3, 10−4, and 10−5), nanoparticle volumetric fraction (φ = 2%, 3%, 4%, and 5%), and the undulations number of the outer wavy cylinder (three cases) on the distributions of isotherms, streamlines, mean Nusselt number (Nuavg), as well as on total entropy production and Bejan number are thoroughly examined. The computational outcomes disclose that dispersing more Cu nanoparticles in the base fluid and creating a flow with higher intensity inside the annulus by raising the Rayleigh number bring about a boosted natural convective flow in the cavity, which improves the heat transmission rate. In addition, it can be noted that owing to the peculiar form of the heated Koch snowflake, nanofluid gets trapped between the angled parts, resulting in uneven temperature profiles with higher values in these places.

Performance evaluation of compact FSS‐integrated flexible monopole antenna for body area communication applications

SummaryThis article presents a compact wearable antenna integrated with a novel frequency selective surface (FSS) made of textile materials. The antenna can be applied in wearable health monitoring applications aimed at the 2.45‐GHz Industrial Scientific and Medical (ISM) band. An array of 4 × 4 octagonal slot‐shaped FSS (106 × 106 mm2) superstrate enhances the performance of the modified Koch curve planar monopole antenna (44.85 × 39 mm2). The Jeans substrate prototype with the overall dimensions of 1.128λ0 × 1.128λ0 × 0.319λ0 covers the bandwidth of 2.45‐GHz ISM band from 2.24 to 2.76 GHz (S11 ≤ −10 dB). The band rejection and reflective nature of the proposed FSS suppress the back radiation and improves the antenna's gain by −20 dB and 6.3 dBi, respectively. The specific absorption rate (SAR) level is reduced to 0.445 W/kg from 24.995 W/kg with a higher reduction rate of 98.21%. Many literature reports produced higher gain and lower SAR but exhibited reduced efficiency when the antenna is placed near the human body. The wearable antenna's frequency response tends to vary under bending conditions for narrowband antennas. Wide‐band monopole antennas produce omnidirectional radiation pattern which tends to affect human body tissues. So stable polarization‐insensitive FSS is placed behind the antenna to enhance the antenna's gain and avoid back radiation. The proposed antenna exhibits stable performance under bending conditions with fractional bandwidth of 22.2%, gain of 8.83 dBi, and produces a better efficiency of 91.30% when the antenna is positioned in the vicinity with human body models. The experimental validation shows that the proposed antenna is a promising candidate for wireless body area networks.

Development of Gas Sensor Based on Fractal Substrate Structures

Gas sensor plays a key role in many applications with sensitivity being a critical performance characteristic. Increasing the surface area of gas sensing material is one approach that can increase sensitivity. Fractal geometries, which have large specific surface area and special fractal dimension, have previously been successfully used in the design of macro and microstructures of gas sensors to improve their performance. In this paper, the influence of geometrical structure of the substrate on the gas sensor performance has been investigated. Two fractal structures (Koch snowflake and Menger sponge) and one traditional structure (Cylinder) were fabricated by 3D printing and coated in Ag:MWCNTs based sensing materials. The fabricated sensors were tested with nitrogen dioxide at different temperatures and humidity. Experimental results show that the sensitivity of gas sensors with fractal structures is increased more than twice that of those with traditional geometrical structures.

Multiband PIFA antenna with Koch Fractal slot for GPS, LTE and WiMAX wireless communication systems

In this article, a five frequency band PIFA antenna for GPS (1.57 GHz), LTE (2.6 GHz) and WiMAX (3.17GHz, 4.1GHz, 4.83GHz) applications is presented. The proposed antenna consists of a PIFA antenna with a Koch fractal slot at the first iteration that was applied to all sides of a square located in the center of the radiating element. The radiating patch modification of the proposed antenna with the fractal slot made the antenna multiband and also contributed to miniaturize the proposed antenna compared to the initial form. The results presented in this work are carried out under the CST MWs software and compared to those obtained by the HFSS software where a good agreement was recorded. The proposed antenna has a gain ranging from 1.13dB to 6.01dB corresponding to the five frequency bands achieved.

An Upside Down Approach to the Koch Curve

An Upside Down Approach to the Koch Curve

Normalizing and classifying shape indexes of cities by ideas from fractals

A standard scientific study comprises two processes: one is to describe how a system works, and the other is to understand why the system works in this way. In order to understand the principle of urban growth, a number of shape indexes are proposed to describe the size and shape of cities. However, the comparability of a shape index is often influenced by the resolution of remote sensing images or digital maps because the calculated values depend on spatial measurement scales. This paper is devoted to exploring the scaling property in classical shape indexes with mathematical methods. Two typical regular fractals, Koch's island and Vicsek's figure, are employed to illustrate the property of scale dependence of many shape indexes. A set of formula are derived from the geometric measure relation to associate the fractal dimension for boundary lines with shape indexes. Based on the ideas of fractals, two main problems are solved in this work. First, several different correlated shape indexes such as circularity ratios and compactness ratios are normalized and unified by substituting Feret's diameter for the major axis of urban figure. Second, three levels of scaling in shape indexes are revealed, and shape indexes are classified into three groups. The significance of this study lies in two aspects. One is helpful for realizing the deep structure of urban form, and the other is useful for clarifying the spheres of application of different types of shape indexes.

Design of a 1×2 CPW Fractal Antenna Array on Plexiglas Substrate with Defected Ground Plane for Telecommunication Applications

In this paper, a fractal antenna array for telecommunication applications is presented. The proposed antenna array is realized on a Plexiglas substrate, has 1×2 radiating elements, and dimensions of 170mm×105mm. The antenna array is composed of two Koch Snowflake patches and is fed by a Coplanar Waveguide (CPW) transmission line. Radiating elements and the ground plane are printed on the top side of the substrate. Defected Ground Structure (DGS) technique is employed to enhance the bandwidth and improve the impedance matching. The proposed antenna array operates at two frequency bands, 1.08-1.32GHz covering the GPS band and 1.7-3.7GHz covering the GSM 1800/1900, UTMS, Bluetooth, LTE, and WiMAX bands. In addition, the antenna has a good performance with efficiency and peak gain of 82% and 6.3dB respectively. These characteristics allow the antenna to be an attractive candidate for telecommunication systems. Design and analysis of different structures were carried out with Ansys HFSS.

Mechanism of Band Gaps in Self-Similar Triangular Lattice With Koch Fractal

Abstract Fractal lattice is a kind of lattices with multifunctional physical characteristics and superior mechanical properties. The wave propagation of the triangular lattice with Koch fractal is calculated by the finite element method and Bloch theorem. The effects of the iteration number on the band gaps and the band edge modes are studied. The finite element software was used to simulate the dynamic response of the triangular lattice with Koch fractal for verifying the vibration suppression performance. The results show that the triangular lattice with Koch fractal can produce multiple and low-frequency band gaps. As an increase of the iteration number, the band gap gradually shifts to a lower frequency. By comparing and analyzing the band edge modes and the eigenmodes of Koch fractal, the mechanisms of the band gaps within the low-frequency ranges are analyzed and discussed in detail. Additionally, the band edge modes exhibit similar vibration modes. Finally, the simulation results of the finite lattice verify the broadband vibration suppression performance of the triangular lattice with Koch fractal. This work provides insights into the lattice dynamic behavior adjusted by Koch fractal, which is beneficial to the periodic lattice for suppressing vibration in engineering applications.

Analysis of Timoshenko beam with Koch snowflake cross-section and variable properties in different boundary conditions using finite element method

This study analyzed a Timoshenko beam with Koch snowflake cross-section in different boundary conditions and for variable properties. The equation of motion was solved by the finite element method and verified by Solidworks simulation in a way that the maximum error was about 2.9% for natural frequencies. Displacement and natural frequency for each case presented and compared to other cases. Significant research achievements illustrate that if we change the Koch snowflake cross-section of the beam from the first iteration to the second, the area and moment of inertia will increase, and we have a 5.2% rise in the first natural frequency. Similarly, by changing the cross-section from the second iteration to the third, a 10.2% growth is observed. Also, the hollow cross-section is considered, which can enlarge the natural frequency by about 26.37% compared to a solid one. Moreover, all the clamped-clamped, hinged-hinged, clamped-free, and free-free boundary conditions have the highest natural frequency for the Timoshenko beam with the third iteration of the Koch snowflake cross-section in solid mode. Finally, examining important physical parameters demonstrates that variable density from a minimum value to the standard value along the beam increases the natural frequencies, while variable elastic modulus decreases it.

Fractal dimension of basin boundaries calculated using the basin entropy

The concept of basin entropy (Sb) was recently introduced as a means to characterize basins of attraction with regard to their complexity. It was also found a connection between Sb and the uncertainty exponent α. This connection allows the calculation of the fractal dimension d of the basin boundary between two basins of attraction. However, this method of calculation has not been explored in the literature. In this work we evaluate the performance of the method based upon the basin entropy in the calculation of the fractal dimension of basin boundaries. For that purpose, the method is applied to the calculation of d for several artificial uniform fractals, such as the Koch island and the Sierpinski Carpet, and the values obtained are compared with the exact dimensions obtained by analytical methods. It is concluded that excellent results are generally obtained if the boxes used in the calculation of Sb are chosen adequately, and a simple criterion for this choice is proposed. Numerical arguments are provided to justify the exclusion of small boxes with low resolution in the calculation of d. While the investigation is motivated by the calculation of d for basin boundaries, the method can be applied to any image containing two distinct regions with a boundary, whose dimension has to be determined.

A remarkable fact for the box dimensions of fractal interpolation curves of R3

In this paper, we give the fractal interpolation curves generated on the Koch curve by Hölder continuous functions and function vertical scaling factors, and at the same time, we present the estimation method of box dimensions of curves of R3 through the estimation of lower and upper box dimensions of obtained curves, where a remarkable fact is that the box dimensions are dependent not only on function vertical scaling factors but also on Hölder exponent of harmonic functions on the Koch curve.

Ultra-wideband bandpass filter with notch band based on quadratic Koch Island structure

An ultra-wideband bandpass filter with a notch band centered at 7.2 GHz is proposed to remove the interference caused by satellite communication signal coexciting within the ultra wide band. The filter comprises of two seperated quadratic koch island structures connected to the main transmission line to generate the notch band at the desired frequency. The designed ultra wide bandpass filter passes frequencies from 3.09 GHz to 10.61 GHz with a notch band from 7.12 to 7.46 GHz centered at 7.2 GHz and with a rejection level of 21.3 dB.The resonant frequency and bandwidth of the notch can be varied by the variation in the physical parameter of the filter. The proposed filter is fabricated, tested and compared with simulated results.