Koch Fractal Research Articles

Quantitative evaluation of eddy current distribution by relative entropy and cross entropy

Koch curve exciting coil of an eddy current probe can adjust the eddy current distributing in more directions at a small domain to enhance the sensitivity of eddy current probe for short defect detection. In this study, a relative entropy and a cross entropy of tangential intersection angle spectrum are proposed to evaluate the eddy current distributions in the different directions when the eddy current probe is positioned at different lift-off distances and excited by different exciting frequency alternative currents. The eddy current distributions induced by a circular and a fractal Koch curve exciting coils are analyzed by the two entropy indices. With the increasing of the lift-off distance or the decreasing of the exciting frequency, the eddy current distributions induced by the Koch curve exciting coil are close to those induced by the circular exciting coil.

Complex and quaternionic Cauchy formulas in Koch snowflakes

ABSTRACT In this paper we derive a Cauchy integral formula for holomorphic and hyperholomorphic functions in domains bounded by a Koch snowflake in two- and three-dimensional setting.

Решение двумерной задачи дифракции волн на фракталоподобных телах методом диаграммных уравнений

The solution of the two-dimensional wave diffraction problem for infinite cylinder of complex cross-section was considered by using the pattern equations method (PEM). A triangle and a Koch snowflake of first iteration were chosen as the geometry of the cross-sections of the cylinder. The numerical algorithms of the PEM for a single scatterer and for a group of bodies with the Dirichlet condition on their boundary are briefly presented, and the results of numerical calculations of the scattering characteristics for the above geometries are obtained using the PEM and the method of continued boundary conditions (MCBC). To check the convergence of the numerical algorithm in both methods, the optical theorem was used. The limits of applicability of the PEM for fractal scatterers are established. It is shown that for all convex bodies the algorithm of the PEM is sufficiently stable and allows obtaining calculation results with an accuracy acceptable in practice. In the case of a non-convex body, namely, a Koch snowflake, the algorithm of the PEM for a single scatterer turns out to be unstable and the acceptable accuracy can be obtained only if this geometry is considered as a group of bodies composed of convex geometries (for example, triangles).

Fractal Design Boosts Extrusion-Based 3D Printing of Bone-Mimicking Radial-Gradient Scaffolds.

Although extrusion-based three-dimensional (EB-3D) printing technique has been widely used in the complex fabrication of bone tissue-engineered scaffolds, a natural bone-like radial-gradient scaffold by this processing method is of huge challenge and still unmet. Inspired by a typical fractal structure of Koch snowflake, for the first time, a fractal-like porous scaffold with a controllable hierarchical gradient in the radial direction is presented via fractal design and then implemented by EB-3D printing. This radial-gradient structure successfully mimics the radially gradual decrease in porosity of natural bone from cancellous bone to cortical bone. First, we create a design-to-fabrication workflow with embedding the graded data on basis of fractal design into digital processing to instruct the extrusion process of fractal-like scaffolds. Further, by a combination of suitable extruded inks, a series of bone-mimicking scaffolds with a 3-iteration fractal-like structure are fabricated to demonstrate their superiority, including radial porosity, mechanical property, and permeability. This study showcases a robust strategy to overcome the limitations of conventional EB-3D printers for the design and fabrication of functionally graded scaffolds, showing great potential in bone tissue engineering.

A Novel Planar Differential Koch Fractal Eddy Current Probe with Parallel Wound Topological Structure

Flexible planar eddy current probes are widely used to detect conductive components with complex surface. In this study, topological transformation is applied to design a differential Koch coil exciting eddy current probe. Two kinds of Koch exciting coils, Koch A and Koch B, were obtained by topological transformation from a three‐dimensional differential exciting eddy current probe. Finite element model simulation is conducted to visualize the differences of eddy current distributions induced by the probes and get the defect signal. A detailed comparison is made among the two kinds of Koch eddy current probes and a circular eddy current probe by experiments. The experiments demonstrate that the sensitivity of the Koch A eddy current probe is higher than that of another two probes for detecting the defect which is shorter than the size of the probe. This work provides a novel method for improving the performance of eddy current probes in the coil structure design.

Modified Koch curve broadband fractal antenna for spectrum sensing in cognitive radio

AbstractThis paper presents the design and development of broadband modified Koch curve microstrip patch antenna with multiband characteristics for spectrum sensing in cognitive radio applications. The proposed spectrum sensing antenna is designed specifically for its efficient operation in the spectrum (1.683–3.05, 4.246–9.714, and 11.25–18 GHz) specified for L, S, C, X, Ku with broadband characteristics and omnidirectional radiation pattern. The validity of the proposed shape is proved by having a comparison with other fractal shapes available in the literature.

Metamaterial-Fractal-Defected Ground Structure Concepts Combining for Highly Miniaturized Triple-Band Antenna Design

Abstract In this paper, a novel method is proposed to increase the gain and radiation efficiency of a compact patch antenna. By employing a combination of three efficient techniques, we have developed a multi-resonance L-DGS antenna with a high gain of 5 dB and an efficiency of 99.6%. Furthermore, a novel compact Double Negative metamaterial unit cell and its equivalent circuit are investigated, to achieve high miniaturization of 30×30 mm2 and multi-band wireless applications (2.8 GHz, 4.1-4.45 GHz, 5.6 GHz). Koch snowflake fractal is introduced along radiation patch edges to improve the antenna matching. The antenna is designed using commercially available package CST software, printed on Rogers RT5880, and the probe feed mechanism is adopted for the antenna excitation. Then, to prove the validation of the antenna design, the equivalent circuit is presented and simulated using ADS of Agilent software. The compared simulation results given by CST, HFSS and ADS software have confirmed the antenna use for WIMAX, C-band and WLAN applications.

On the Classification of Fractal and Non-Fractal Objects in Two-Dimensional Space

The method of numerical Fourier analysis is used to investigate the fractal properties of 2D objects with micrometer–centimeter sizes. This numerical method simulates the small-angle light scattering experiment. Different geometric 2D-regular fractals, such as the Sierpinski carpet, Sierpinski triangle, Koch snowflake, and Vishek snowflake, are studied. We can divide 2D fractals, by analogy with 3D fractals, into “plane” and “boundary” fractals with fractal dimensions lying in the intervals from 1 to 2 and from 2 to 3, respectively. For an object with a smooth boundary, i.e., a circle, the model small-angle scattering curve decreases according to the power law q–3, where q is the momentum transferred.

PolySimp: A Tool for Polygon Simplification Based on the Underlying Scaling Hierarchy

Map generalization is a process of reducing the contents of a map or data to properly show a geographic feature(s) at a smaller extent. Over the past few years, the fractal way of thinking has emerged as a new paradigm for map generalization. A geographic feature can be deemed as a fractal given the perspective of scaling, as its rough, irregular, and unsmooth shape inherently holds a striking scaling hierarchy of far more small elements than large ones. The pattern of far more small things than large ones is a de facto heavy tailed distribution. In this paper, we apply the scaling hierarchy for map generalization to polygonal features. To do this, we firstly revisit the scaling hierarchy of a classic fractal: the Koch Snowflake. We then review previous work that used the Douglas–Peuker algorithm, which identifies characteristic points on a line to derive three types of measures that are long-tailed distributed: the baseline length (d), the perpendicular distance to the baseline (x), and the area formed by x and d (area). More importantly, we extend the usage of the three measures to other most popular cartographical generalization methods; i.e., the bend simplify method, Visvalingam–Whyatt method, and hierarchical decomposition method, each of which decomposes any polygon into a set of bends, triangles, or convex hulls as basic geometric units for simplification. The different levels of details of the polygon can then be derived by recursively selecting the head part of geometric units and omitting the tail part using head/tail breaks, which is a new classification scheme for data with a heavy-tailed distribution. Since there are currently few tools with which to readily conduct the polygon simplification from such a fractal perspective, we have developed PolySimp, a tool that integrates the mentioned four algorithms for polygon simplification based on its underlying scaling hierarchy. The British coastline was selected to demonstrate the tool’s usefulness. The developed tool can be expected to showcase the applicability of fractal way of thinking and contribute to the development of map generalization.

A Second-Iteration Square Koch Fractal Slot Antenna for UHF Downlink Telemetry Applications in CubeSat Small Satellites

This article presents the design, fabrication, and measurement of a square Koch fractal slot antenna for UHF band using both the FR4-G10 and Cuclad 250 substrates. Conveniently, this 56.56 cm full-length antenna possesses a geometry that allows it to be incorporated into the standardized 10 cm × 10 cm faces of the CubeSats. Furthermore, it is shown that both selected substrates exhibit an acceptable performance at the frequency of interest despite the economic cost difference and relative permittivity. Hence, the commercial FR4-G10 antenna substrate can be preferred because of its low-cost and admissible performance at 458 MHz, which is a frequency in the UHF band that is commonly used for telemetry, tracking, and command downlinks of CubeSats. Measurements show that the proposed antenna exhibits a reflection coefficient of −16.53 dB, a bandwidth of 22.62 MHz at −10 dB, a VSWR of 1.3508, a normalized impedance of 0.794 − j0.173 at 50 Ω, and a directivity of 2.24 dBi. The contribution of this work consists in the use of a fractal geometry to construct a low-cost slot antenna working at UHF frequencies over the limited area of the CubeSat faces and in order to optimize the area for an eventual coexistence with solar cells.

Journey to the moon and to the sun on the Koch Curve

ABSTRACT The article shows that even a small number of iterations give a very large length of the Koch Curve. The overall purpose of this work is to show that even a small number of iterations can give a very complex fractal structure. In this case it is the very big length of Koch's curve. As examples, the distance from the Earth to the Moon and from the Earth to the Sun was compared to the length of the Koch curve.

A metamaterial loaded hybrid fractal multiband antenna for wireless applications with frequency band reconfigurability characteristics

Abstract In the present article authors propose the design and analysis of an octagonal shape multiband metamaterial loaded antenna with implementation of hybrid fractal geometry for wireless applications. Multiband features in the antenna structure is realized by applying the slotted and hybrid fractalization of Moore and Koch curve approach in radiating section along with introduction of two metamaterial SRR cells. The frequency band reconfigurability characteristics in proposed design is achieved by placing the PIN diode inside the connecting strip between the central hybrid fractal geometry and feedline. During forward bias condition of PIN diode antenna structure resonates at hepta (seven) band mode at WiMAX (3.5 GHz)/Lower C-band (4.41 GHz)/WLAN (5.4/5.8 GHz)/Lower X-band (8.26 GHz)/Upper X-band (10.48 GHz)/Lower Ku-band (13.35 GHz)/Middle Ku-band (14.42 GHz) wireless standards with S 11 ≤ −10 dB. Proposed antenna represent the hexa and hepta band features during reverse bias (OFF-state) and forward bias condition (ON-state) of PIN diode respectively. A stable and consistent radiation patterns, appropriate impedance matching and an acceptable gain are achieved at all the operating frequencies of the proposed antenna.

Compact Circular CSRR Loaded Koch Fractal Inspired Frequency Band Reconfigurable Antenna

A compact frequency band reconfigurable dual-mode antenna utilizing both Koch fractal geometry and circular complementary split ring resonators (CSRRs) is presented in this paper. The proposed antenna can be electrically switched into both dual- and single-band resonance modes. A Koch curve derivative novel fractal curve is used to design the patch boundary of the antenna. Two optimized circular CSRRs are loaded inside the antenna geometry to achieve double-band resonance. The antenna operates at both 5.1[Formula: see text]GHz (upper WLAN band) and 9.6[Formula: see text]GHz ([Formula: see text]-band) in dual resonance mode and resonates only at 8[Formula: see text]GHz ([Formula: see text]-band) in single resonance mode. A properly biased RF-range PIN diode placed at the slotted ground plane reconfigures the antenna in two operating modes by switching into its ON–OFF states. The antenna is fabricated on cost-efficient epoxy (FR-4) substrate consuming the maximum dimension of [Formula: see text] which is much more compact than the conventional rectangular patch antennas designed for similar bands. Stable radiation characteristics with reduced cross-polarization are achieved at all the resonant bands of the antenna. A prototype of the proposed antenna is fabricated and measured. The simulated results are in good agreement with the measured ones.

A nonparabolic conduction band study of circular quantum dot optical properties: modeling of surface roughness by using Koch snowflakes

In this work, we investigate the optical properties of single and multiple quantum dots (QDs) based on AlxGa1-xAs/GaAs with two Koch snowflake-shaped and circular geometries. The idea is to determine the impact of the surface roughness of the circular QDs on the mentioned optical properties. This is an important issue in designing optoelectronic devices (e.g., photodetectors or light-emitting systems) because the position and magnitude of the absorption coefficient play important roles in devising them. We have also studied the effects of the conduction band nonparabolicity, QD size, number of QDs, and composition parameter x on the absorption coefficient. For this purpose, we have used an efficient finite difference method to solve the two-dimensional Schrodinger equation. The computational algorithm of the Koch snowflake-shaped QD production is also described.

The area of a rough black hole

We investigate the consequences for the black hole area of introducing fractal structure for the horizon geometry. We create a three-dimensional spherical analogue of a ‘Koch Snowflake’ using a infinite diminishing hierarchy of touching spheres around the Schwarzschild event horizon. We can create a fractal structure for the horizon with finite volume and infinite (or finite) area. This is a toy model for the possible effects of quantum gravitational spacetime foam, with significant implications for assessments of the entropy of black holes and the universe, which is generally larger than in standard picture of black hole structure and thermodynamics, potentially by very considerable factors. The entropy of the observable universe today becomes S≈10120(1+Δ/2), where 0≤Δ≤1, with Δ=0 for a smooth spacetime structure and Δ=1 for the most intricate. The Hawking lifetime of black holes is also reduced.

Elastic Wave Propagation in Lattice Metamaterials with Koch Fractal

In this study, the wave propagation properties of lattice metamaterials with Koch fractal structures are investigated in terms of band structures and directional wave propagation. The analytical models of lattice metamaterials are established using the finite element method, and the dispersion relation is solved using the Bloch’s theorem. The band structures of the lattice metamaterials with different numbers of iterations are studied, and the group velocities at a selected frequency are calculated to analyze the directional wave propagation characteristics. Furthermore, dynamic responses of the finite structures are calculated using commercial finite element software to verify the band gaps and directional wave propagation behaviors in the lattice metamaterials. The results show that multiple and low band gaps are present in the lattice materials with various geometric parameters of the Koch fractal, and the position of the lowest band gap decreases as the number of iterations increases. The results indicate the potential applications of lattice metamaterials with Koch fractals for vibration isolation and multi-functional design.

Abstract similarity, fractals and chaos

<p style='text-indent:20px;'>A new mathematical concept of abstract similarity is introduced and is illustrated in the space of infinite strings on a finite number of symbols. The problem of chaos presence for the Sierpinski fractals, Koch curve, as well as Cantor set is solved by considering a natural similarity map. This is accomplished for Poincaré, Li-Yorke and Devaney chaos, including multi-dimensional cases. Original numerical simulations illustrating the results are presented.

Implementation of hybrid fractal metamaterial inspired frequency band reconfigurable multiband antenna for wireless applications

In the present paper authors propose the design and analysis of a hepta band metamaterial inspired octagonal shape antenna using hybrid fractals for wireless applications. Multiband characteristics in the designed antenna is accomplished by introducing of slotted octagonal shape radiating part with hybrid fractal form of Moore curve and Koch curve and two SRR cells. The frequency band reconfigurability is obtained with aid of PIN diodes placed inside the strips connected between Moore curve (fused with centered Koch curve) and feedline. During ON mode of PIN diode antenna operates at seven microwave frequency S-band WiMAX (3.4~3.69 GHz—IEEE 802.16e)/Lower C-band terrestrial fixed and mobile broadband application (4.25~4.76 GHz)/C-band WLAN (5.15~5.35/5.75~5.825 GHz—IEEE 802.11a] (5.4~5.9 GHz)/Lower X-band Earth exploration-satellite service ITU region 2 (7.9~8.4 GHz)/Upper X-band Amateur satellite operating band (10.45~10.50 GHz)/Lower Ku-band Radar communication application (13.25~13.75 GHz)/Middle Ku-band Geostationary satellite service (14.2~14.5 GHz) covering various wireless applications. Proposed design exhibit hexa/hepta band features during OFF/ON mode of PIN diode. An acceptable gain, stable radiation characteristics, and good impedance matching are observed at all the resonant frequencies of the proposed structure.

Koch Fractal in Non-Euclidean Geometries

Koch Fractal in Non-Euclidean Geometries

Meet fractal curves with 1C:MathKit

The article presents a methodological approach to studying iterative processes in the school course of geometry, by the example of constructing a Koch snowflake fractal curve and calculating a few characteristics of it. The interactive creative environment 1C:MathKit is chosen to visualize the method discussed. By performing repetitive constructions and algebraic calculations using ICT tools, students acquire a steady skill of work with geometric objects of various levels of complexity, comprehend the possibilities of mathematical interpretation of iterative processes in practice, and learn how to understand the dialectical unity between finite and infinite parameters of flat geometric figures. When students are getting familiar with such contradictory concepts and categories, that replenishes their experience of worldview comprehension of the subject areas they study through the concept of “big ideas”. The latter allows them to take a fresh look at the processes in the world around. The article is a matter of interest to schoolteachers of computer science and mathematics, as well as university scholars who teach the course “Concepts of modern natural sciences”.