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Journal articles on the topic 'Fractal Koch geometry'

Author: Grafiati

Published: 4 June 2025

Last updated: 1 August 2025

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1

Camp, Dane R. "A Fractal Excursion." Mathematics Teacher 84, no. 4 (1991): 265–75. http://dx.doi.org/10.5951/mt.84.4.0265.

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Recently, chaos theory and the related topic of fractal geometry have blossomed as creative fields of study in mathematics and physics. Fractals are shapes containing self-similarity on arbitrary magnification. One such object, the Koch curve, is generated by simple recursion on an equilateral triangle. The process used to produce the curve is a great way to introduce students to some concepts of fractal geometry.

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Jamil, Atif, Muhammad Rauf, Abdul Sami, Arsalan Ansari, and Muhammad Dawood Idrees. "A Wideband Hybrid Fractal Ring Antenna for WLAN Applications." International Journal of Antennas and Propagation 2022 (February 21, 2022): 1–8. http://dx.doi.org/10.1155/2022/6136916.

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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

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Jaffri, Zain Ul Abidin, Zeeshan Ahmad, Asif Kabir, and Syed Sabahat Hussain Bukhari. "A novel miniaturized Koch-Minkowski hybrid fractal antenna." Microelectronics International 39, no. 1 (2021): 22–37. http://dx.doi.org/10.1108/mi-07-2021-0069.

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Purpose Antenna miniaturization, multiband operation and wider operational bandwidth are vital to achieve optimal design for modern wireless communication devices. Using fractal geometries is recognized as one of the most promising solutions to attain these characteristics. The purpose of this paper is to present a unique structure of patch antenna using hybrid fractal technique to enhance the performance characteristics for various wireless applications and to achieve better miniaturization. Design/methodology/approach In this paper, the authors propose a novel hybrid fractal antenna by combi

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TATOM, FRANK B. "THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS." Fractals 03, no. 01 (1995): 217–29. http://dx.doi.org/10.1142/s0218348x95000175.

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The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-dif

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Bigerelle, Maxence, François Berkmans, and Julie Lemesle. "Evaluating the Fractal Pattern of the Von Koch Island Using Richardson’s Method." Fractal and Fractional 9, no. 8 (2025): 483. https://doi.org/10.3390/fractalfract9080483.

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The principles of fractal geometry have revolutionized the characterization of complex geometric objects since Benoit Mandelbrot’s groundbreaking work. Richardson’s method for determining the fractal dimension of boundaries laid the groundwork for Mandelbrot’s later developments in fractal theory. Despite extensive research, challenges remain in accurately calculating fractal dimensions, particularly when dealing with digital images and their inherent limitations. This study examines the numerical artifacts introduced by Richardson’s method when applied to the Von Koch Island, a classic fracta

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Mathur, Vinita, and Dr Manisha Gupta. "Morphology of Koch Fractal Antenna." INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY 13, no. 2 (2014): 4157–63. http://dx.doi.org/10.24297/ijct.v13i2.2902.

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Antenna is paramount element for the radio communication entity using radio frequency and microwaves.Â In twenty-first century wireless communication systems, there is a demand for wider bandwidth, multiband and low profile antennas for both commercial and military purposes. This has initiated antenna analysis in various directions; one of them is using fractal shaped antenna elements. Fractal concepts have emerged and advanced as a unique technique for designing compact UWB antennas, because of the self-similarityand space-filling attributes. In a fractal antenna, the multiple frequencies of

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Zhao, Shanrong, Jin Tan, Jiyang Wang, Xiaohong Xu, and Hong Liu. "A Dendrite with "Sierpinski Gasket" Fractal Morphology in Matt Glaze of LiAlSiO4-SiO2 System." Fractals 11, no. 03 (2003): 271–76. http://dx.doi.org/10.1142/s0218348x03001525.

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In this paper, we introduce a dendritic crystal, formed in matt glaze of LiAlSiO 4- SiO 2, having "Sierpinski gasket" fractal morphology. The crystal structure of this "Sierpinski gasket" dendrite is β-quartz. β-quartz can grow two kinds of fractal patterns: snow-shaped dendrite and "Sierpinski gasket" dendrite, depending on different supercooling conditions. These two kinds of fractals can develop together in one dendritic crystal. The evolution of the boundary morphologies between these two kinds of fractal dendrites can be described by another fractal — Koch curve. The "Sierpinski gasket" d

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Purnomo, Kosala Dwidja, Siti Fatimah, and Bagus Juliyanto. "Generation of Fractal Objects with Iterated Function System on the Developments of Trellis Ornament Designs." BERKALA SAINSTEK 13, no. 1 (2025): 1–7. https://doi.org/10.19184/bst.v13i1.25656.

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Fractals are one of a mathematical concept that provides artistic value and is therefore widely used to design various kinds of objects. The purpose of this study is to obtain various trellis ornament designs generated from fractal objects. Some fractal objects that will be used are Koch Snowflake (m,n,c), Koch Anti-Snowflake (m,n,c) and dragon curve. The basic trellis pattern is built from basic geometry, namely line segments, rhombuses and elliptical curved lines with certain sizes. In this study, the generation of fractal objects was carried out using the Iterated Function Systems (IFS) met

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RAMÍREZ, JOSÉ L., GUSTAVO N. RUBIANO, and BORUT JURČIČ ZLOBEC. "GENERATING FRACTAL PATTERNS BY USING p-CIRCLE INVERSION." Fractals 23, no. 04 (2015): 1550047. http://dx.doi.org/10.1142/s0218348x15500474.

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In this paper, we introduce the [Formula: see text]-circle inversion which generalizes the classical inversion with respect to a circle ([Formula: see text]) and the taxicab inversion [Formula: see text]. We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the [Formula: see text]-circle inversion.

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LI, JIE, YONGPING HUANG, CHENGBIN ZHANG, and XIANGDONG LIU. "NUMERICAL STUDY ON THE SOLIDIFICATION PERFORMANCE OF A LATENT HEAT STORAGE UNIT WITH KOCH-FRACTAL FIN." Fractals 27, no. 07 (2019): 1950108. http://dx.doi.org/10.1142/s0218348x19501081.

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Inspired by the snowflake structure, an innovative Koch-fractal fin is proposed to optimize the fin geometry of the latent heat thermal energy storage (LHTES) units. A model of unsteady heat transfer accompanied with phase change is developed and numerically analyzed to investigate the effect of fin structure on the discharging process of a LHTES unit. The dynamic response of heat release, the dynamic temperature response and solidification front evolution of a LHTES unit with Koch-fractal fins are discussed and compared with the corresponding radial fins. Furthermore, a comprehensive evaluati

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ZHANG, SHUAI, XUEYE CHEN, ZHONGLI WU, and YUE ZHENG. "NUMERICAL STUDY ON KOCH FRACTAL BAFFLE MICROMIXER." Fractals 27, no. 03 (2019): 1950026. http://dx.doi.org/10.1142/s0218348x19500269.

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This paper is mainly to study the application of Koch fractal baffle to passive micromixers. It can be determined that the mixing efficiency of secondary Koch fractal baffle (SKFB) micromixer is better than that of primary Koch fractal baffle (PKFB). We compare and analyze the mixing efficiency when the angle between the baffle and the microchannel is [Formula: see text], [Formula: see text] and [Formula: see text] with the height 100[Formula: see text][Formula: see text]m. With the changing of the angle, it contributes to enhance the chaotic convection of the micromixer. Especially at the ang

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CHEN, YAN-GUANG. "FRACTAL TEXTURE AND STRUCTURE OF CENTRAL PLACE SYSTEMS." Fractals 28, no. 01 (2020): 2050008. http://dx.doi.org/10.1142/s0218348x20500085.

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The boundaries of central place models proved to be fractal lines, which compose fractal texture of central place networks. However, the fractal texture cannot be verified by empirical analyses based on observed data. On the other hand, fractal structure of central place systems in the real world can be empirically confirmed by positive studies, but there are no corresponding models. The spatial structure of classic central place models bears Euclidean dimension [Formula: see text] rather than fractal dimensions [Formula: see text]. This paper is devoted to deriving structural fractals of cent

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Jahanbakht, Mohammad, and Abbas Ali Lotfi-Neyestanak. "COPLANAR MEMS PHASED ARRAY ANTENNA USING KOCH FRACTAL GEOMETRY." Progress In Electromagnetics Research M 17 (2011): 29–42. http://dx.doi.org/10.2528/pierm11010301.

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Morillo, Pablo Lupera, Gary Flores Cadena, and Ricardo Merizalde. "Design and Testing of Fractal Antenna Parameters based on the Koch Curve for Reception of Digital Terrestrial Television Signals in the UHF Band." Law, State and Telecommunications Review 11, no. 1 (2019): 159–72. http://dx.doi.org/10.26512/lstr.v11i1.24855.

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Purpose – In this research paper, the electrical and radiation characteristics of a proposed fractal antenna based on the Koch curve in the second iteration for reception of digital terrestrial television signals are designed and analyzed by laboratory tests. Methodology/approach/design – The design is based on the concepts of fractal geometry and on a previously designed antenna, which is adapted to obtain a different frequency of operation; the designed antenna is constructed in three different ways, finally, they are tested in the lab using vector-network-analyzer, that allows to meas

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VINOY, K. J., JOSE K. ABRAHAM, and V. K. VARADAN. "IMPACT OF FRACTAL DIMENSION IN THE DESIGN OF MULTI-RESONANT FRACTAL ANTENNAS." Fractals 12, no. 01 (2004): 55–66. http://dx.doi.org/10.1142/s0218348x04002288.

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During the last few decades, fractal geometries have found numerous applications in several fields of science and engineering such as geology, atmospheric sciences, forest sciences, physiology and electromagnetics. Although the very fractal nature of these geometries have been the impetus for their application in many of these areas, a direct quantifiable link between a fractal property such as dimension and antenna characteristics has been elusive thus far. In this paper, the variations in the input characteristics of multi-resonant antennas based on generalizations of Koch curves and fractal

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SENGUPTA, KAUSHIK, and K. J. VINOY. "A NEW MEASURE OF LACUNARITY FOR GENERALIZED FRACTALS AND ITS IMPACT IN THE ELECTROMAGNETIC BEHAVIOR OF KOCH DIPOLE ANTENNAS." Fractals 14, no. 04 (2006): 271–82. http://dx.doi.org/10.1142/s0218348x06003313.

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In recent years, fractal geometries have been explored in various branches of science and engineering. In antenna engineering several of these geometries have been studied due to their purported potential of realizing multi-resonant antennas. Although due to the complex nature of fractals most of these previous studies were experimental, there have been some analytical investigations on the performance of the antennas using them. One such analytical attempt was aimed at quantitatively relating fractal dimension with antenna characteristics within a single fractal set. It is however desirable t

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TSIANOS, KONSTANTINOS I., and RON GOLDMAN. "BEZIER AND B-SPLINE CURVES WITH KNOTS IN THE COMPLEX PLANE." Fractals 19, no. 01 (2011): 67–86. http://dx.doi.org/10.1142/s0218348x11005221.

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We extend some well known algorithms for planar Bezier and B-spline curves, including the de Casteljau subdivision algorithm for Bezier curves and several standard knot insertion procedures (Boehm's algorithm, the Oslo algorithm, and Schaefer's algorithm) for B-splines, from the real numbers to the complex domain. We then show how to apply these polynomial and piecewise polynomial algorithms in a complex variable to generate many well known fractal shapes such as the Sierpinski gasket, the Koch curve, and the C-curve. Thus these fractals also have Bezier and B-spline representations, albeit in

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Kumar, Tarun, and Hamid Ali. "A Parabolic Planer Monopole Fractal Antenna with Koch." International Journal for Research in Applied Science and Engineering Technology 10, no. 10 (2022): 88–93. http://dx.doi.org/10.22214/ijraset.2022.46941.

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Abstract: Modern telecommunication systems require antenna with wider bandwidth and smaller dimensions. Various antennas for wideband operation have been studied for communication and radar system. The fractal antennas are preferred due to small size, light weight and easy installation. In this paper a parabolic planer monopole antenna using the Koch based fractal geometry is introduced for multiband operation. The design achieves a good input impedance match and linear phase of S11 throughout the pass band (1—3.5 GHz and -10dB criterion for impedance bandwidth). The use of fractal pattern in

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Abdyraeva, N. "Feature of wireless broadband communication LTE using a fractal antenna based on Koch curve." Bulletin of Science and Practice 4, no. 3 (2018): 164–69. https://doi.org/10.5281/zenodo.1198684.

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Fractal antenna is an antenna, which active part has the form of a self-similar curve, andalso a curve similar to the dividing or consisting of similar segments of the figure. In order to have an idea of his form, it is necessary to study fractal geometry. In this article considered the features of the use of fractal antennas, in particular, antennas based on the curve Koch, for wireless broadband access systems. For this type of antenna were given, the basic electrodynamic characteristics. The fractal antenna was simulated in the MMANAGAL system, where it is possible to create, edit, analyze

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Korchazhkina, O. M. "Meet fractal curves with 1C:MathKit." Informatics and education, no. 3 (May 13, 2020): 38–48. http://dx.doi.org/10.32517/0234-0453-2020-35-3-38-48.

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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 ho

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Tripathi, Shrivishal, Akhilesh Mohan, and Sandeep Yadav. "Hexagonal fractal ultra‐wideband antenna using Koch geometry with bandwidth enhancement." IET Microwaves, Antennas & Propagation 8, no. 15 (2014): 1445–50. http://dx.doi.org/10.1049/iet-map.2014.0326.

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Kassym, Ruslan, Tansaule Serikov, Begayda Sarsekulov, and Nursultan Yernazarov. "DEVELOPMENT OF KOCH'S MINIATURE HEXAGONAL FRACTAL ANTENNA FOR 5G AND IOT APPLICATIONS." Вестник Алматинского университета энергетики и связи 4, no. 67 (2024): 86–96. https://doi.org/10.51775/2790-0886_2024_67_4_86.

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This study focuses on a compact hexagonal broadband antenna optimized for 5G millimeter-wave applications using Koch fractals to enhance performance. The research addresses the challenges of achieving high frequencies, expanding bandwidth, and reducing high-frequency interference, all critical for efficient 5G network operation. The antenna achieved a maximum gain of 5 dB at 27.58 GHz, with an impressive bandwidth range from 26.5 to 40 GHz. Koch fractal geometry and defective ground planes significantly improved impedance matching and broadened the bandwidth, offering superior performance over

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Zhu, Yuwen, Junjie Deng, Wei Xiong, Tianyu You, and Wei Zhou. "Koch Hierarchical Honeycomb: A Fractal-Based Design for Enhanced Mechanical Performance and Energy Absorption." Materials 16, no. 10 (2023): 3670. http://dx.doi.org/10.3390/ma16103670.

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A novel energy-absorbing structure, the Koch hierarchical honeycomb, which combines the Koch geometry with a conventional honeycomb structure, is proposed in this work. Adopting a hierarchical design concept using Koch has improved the novel structure more than the honeycomb. The mechanical properties of this novel structure under impact loading are studied by finite element simulation and compared with the conventional honeycomb structure. To effectively verify the reliability of the simulation analysis, quasi-static compression experiments were conducted on 3D-printed specimens. The results

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ALLOUCHE, J. P., and G. SKORDEV. "VON KOCH AND THUE-MORSE REVISITED." Fractals 15, no. 04 (2007): 405–9. http://dx.doi.org/10.1142/s0218348x07003630.

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We revisit the relation between the von Koch curve and the Thue-Morse sequence given in a recent paper of Ma and Goldener by relating their study to papers written by Coquet and Dekking at the beginning of the 1980s. We also emphasize that more general links between fractal objects and automatic sequences can be found in the literature.

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Al-Janabi, Israa Mohsin Kadhim, Zahraa Ahmed AL-Mammori, Sabah Mohammed Abd Mosehab, et al. "An Effective Method for Compute the Roughness of Fractal Facades Based on Box-Counting Dimension (D_b)." BIO Web of Conferences 97 (2024): 00037. http://dx.doi.org/10.1051/bioconf/20249700037.

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Benoit Mandelbrot coined the word “fractal” in the late 1970s, but an object is now defined as fractals in form known to artists and mathematicians for centuries. A fractal object is self-similar in that the subsections of the object are somewhat similar to the whole object. No matter how small the subdivision is, the subsection contains no less detail than the whole. Atypical example of a fractal body is the “snowflake curve” (invented by Helga von Koch (1870-1924) in 1904. There are as many relationships between architecture, the arts, and mathematics as symmetry. The golden ratio, the Fibon

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Salmingo, Rene. "Development of Printed Log-Periodic Planar Antenna using Fractal Kochgeometry." Research Journal of Education, Science and Technology 1, no. 1 (2021): 1–12. https://doi.org/10.63179/rjest.v1i1.4.

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Digital terrestrial television (TV) broadcasting has been adopted in the Philippines with the Digital Switch last 2017. TV viewers can now experience the digital picture and sound through free-to-air TV signals in the household. Nevertheless, the quality of TV pictures and sound studies on digital TV has not been realized because it requires the International Telecommunication Union (ITU) assessment (ITU-R BT.500-13) reference. This paper aimed to develop a printed log[1]periodic planar antenna where the antenna arrays were based on Fractal Koch geometry and determine the digital picture and s

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Bespalov, Andrey N., Alexander L. Buzov, Dmitry S. Klyuev, and Anatoly M. Neshcheret. "Research of antenna complexes using chiral metamaterials and fractal geometry of radiators for MIMO systems." Physics of Wave Processes and Radio Systems 23, no. 4 (2021): 97–110. http://dx.doi.org/10.18469/1810-3189.2020.23.4.97-110.

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This article is devoted to the study of the possibilities of increasing spectral efficiency in MIMO systems by using antennas with substrates of biisotropic and bianisotropic chiral metamaterials and various types of fractal emitters, in particular, fractal structures in the form of a Sierpinski triangle, Koch and Gilbert curves, as well as a dipole triangular antenna of complex configuration FRM. The spectral efficiency was calculated by using one of the variations of the Shannon formula, which includes a complete matrix of Z-parameters. In turn, this matrix was determined using the software

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Chen, Guolong, Weimin Zhang, and Weihan Pang. "Koch curve fractal geometry excitation probe for eddy current non-destructive testing." Measurement 124 (August 2018): 470–78. http://dx.doi.org/10.1016/j.measurement.2018.04.031.

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Blair, Macdonald. "The Fractal Corresponds with Light and Foundational Quantum Problems." International Journal of Quantum Foundations 5, no. 2 (2023): 13–27. https://doi.org/10.5281/zenodo.7938263.

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After nearly one hundred years after its origins, foundational quantum mechanics remains one of the greatest unexplained mysteries in physicists today. Within this time, chaos theory and its geometry— the fractal—has developed. In this paper, the propagation behaviour of a simple iterating fractal—the Koch Snowflake—was described, analysed and discussed. From an arbitrary observation point within the fractal set the fractal propagates forward by oscillation, and retrospectively— viewing behind—it grows exponentially from a point beginning. The fractal propag

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Mathur, Rajeev, Sunil Joshi, and Krishna C. Roy. "A Novel Multiband Koch Loop Antenna Using Fractal Geometry for Wireless Communication System." International Journal of Wireless & Mobile Networks 3, no. 5 (2011): 161–72. http://dx.doi.org/10.5121/ijwmn.2011.3513.

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Davydov, Aleksey S., Dmitry B. Demin, and Dmitry V. Krysanov. "SOLUTION OF THE TWO-DIMENSIONAL WAVE DIFFRACTION PROBLEM ON FRACTAL BODIES BY THE PATTERN EQUATIONS METHOD." T-Comm 15, no. 9 (2021): 4–10. http://dx.doi.org/10.36724/2072-8735-2021-15-9-4-10.

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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 bou

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Mistry, Dhiren, James R. Dawson, and Alan R. Kerstein. "The multi-scale geometry of the near field in an axisymmetric jet." Journal of Fluid Mechanics 838 (January 18, 2018): 501–15. http://dx.doi.org/10.1017/jfm.2017.899.

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A characteristic feature of axisymmetric jets, and turbulent shear flows in general, is the entrainment of mass across the turbulent/non-turbulent interface (TNTI). The multi-scale nature of the TNTI surface area was recently observed to exhibit power-law scaling with a fractal dimension, $D_{f}$, between $D_{f}=2.3{-}2.4$, inferred from two-dimensional data, in both high Reynolds number boundary layers and the far field of axisymmetric jets. In this paper, we show that the fractal scaling previously observed in the far field of an axisymmetric jet is established at the end of the potential co

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Chairunnisa, Chairunnisa, Devy Freshia Sihaloho, and Achmad Munir. "Size Reduction of Printed Log-peri odic Dipole Array Antenna Using Fractal Koch Geometry." International Journal on Electrical Engineering and Informatics 7, no. 2 (2015): 226–36. http://dx.doi.org/10.15676/ijeei.2015.7.2.5.

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Simon, Jorge, Jorge Flores-Troncoso, Jose Luis Alvarez-Flores, et al. "A Second-Iteration Square Koch Fractal Slot Antenna for UHF Downlink Telemetry Applications in CubeSat Small Satellites." International Journal of Antennas and Propagation 2020 (October 9, 2020): 1–10. http://dx.doi.org/10.1155/2020/9672959.

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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

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SUN, YU, JIAHUI ZOU, MEIFENG DAI, XIAOQIAN WANG, HUALONG TANG, and WEIYI SU. "EIGENTIME IDENTITY OF THE WEIGHTED KOCH NETWORKS." Fractals 26, no. 03 (2018): 1850042. http://dx.doi.org/10.1142/s0218348x18500421.

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The eigenvalues of the transition matrix of a weighted network provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to biased walks. Although various dynamical processes have been investigated in weighted networks, analytical research about eigentime identity on such networks is much less. In this paper, we study analytically the scaling of eigentime identity for weight-dependent walk on small-world networks. Firstly, we map the classical Koch fractal to a network, called Koch network. According to the proposed mapping, we pr

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PARVATE, ABHAY, SEEMA SATIN, and A. D. GANGAL. "CALCULUS ON FRACTAL CURVES IN Rn." Fractals 19, no. 01 (2011): 15–27. http://dx.doi.org/10.1142/s0218348x1100518x.

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A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called Fα-integral, where α is the dimension of F. A derivative along the fractal curve called Fα-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize its algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between th

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Saraswat, Ritesh Kumar, and Mithilesh Kumar. "A metamaterial loaded hybrid fractal multiband antenna for wireless applications with frequency band reconfigurability characteristics." Frequenz 74, no. 11-12 (2020): 401–16. http://dx.doi.org/10.1515/freq-2020-0022.

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AbstractIn 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

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CAMP, DANE R. "FLIPPING COINS, FOLDING PAPER, AND FINDING FAMILIAR FRACTALS IN THE TOWER OF HANOI." Fractals 17, no. 01 (2009): 83–89. http://dx.doi.org/10.1142/s0218348x09004193.

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This manuscript describes three activities connecting the Tower of Hanoi puzzle to three familiar fractal forms. The first connects coin flipping, paper folding, and the Tower of Hanoi to the Dragon Curve. The second illustrates mathematician Ian Stewart's method for showing how the relationships between possible states of the Tower of Hanoi are related to stages in Sierpinski's Gasket. The final activity compares right-left moves of Tower of Hanoi disks to iterations of the Von Koch Curve.

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Kassym, Ruslan, Turdybek Balgynbek, Tansaule Serikov, et al. "Development a novel design of miniaturized heptagonal Koch fractal wide band antenna for 5G mm wave and IoT applications." Eastern-European Journal of Enterprise Technologies 3, no. 5 (129) (2024): 6–14. http://dx.doi.org/10.15587/1729-4061.2024.306712.

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The object of the study is a compact heptagonal broadband antenna specially designed for use in the 5G millimeter band using Koch fractals to improve performance. As a result of the study, the most important problem of achieving higher gain, improving bandwidth and reducing interference at higher frequencies, which is necessary for the effective functioning of 5G networks, was solved. As a result, the maximum realized gain of 5 dB was obtained at a frequency of 27.58 GHz with an impressive bandwidth in the range from 26.5 to 40 GHz. The use of Koch fractal geometry and defective ground planes

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N., S. M. Yaziz, K. A. Rahim M., Zubir F., S. Nadzir N., Dewan R., and Dewan R. "The Improvement of first Iteration Log Periodic Fractal Koch Antenna with Slot Implementation." International Journal of Electrical and Computer Engineering (IJECE) 8, no. 4 (2018): 2564–70. https://doi.org/10.11591/ijece.v8i4.pp2564-2570.

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In this paper, a slotted is introduced at each of the radiating elements on the 1<sup>st</sup> iteration log periodic fractal Koch antenna (LPFKA). The antenna is designed to testify the appropriate performance at UHF Digital television which operates from 4.0 GHz to 1.0 GHz. The dimension of the conventional 0<sup>th</sup> iteration LPKFA is successfully reduced by 17% with the implementation of slotted. The results show a good agreement with a stable radiation pattern across the operating bandwidth, stable gain more than 5 dBi and reflection coefficient of below -10 dB over the desired frequ

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SANTRA, S. B., and B. SAPOVAL. "MULTIFRACTAL COLLISION SPECTRUM OF BALLISTIC PARTICLES WITH FRACTAL SURFACES." Fractals 13, no. 01 (2005): 9–18. http://dx.doi.org/10.1142/s0218348x05002751.

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Ballistic particles interacting with irregular surfaces are representative of many physical problems in the Knudsen diffusion regime. In this paper, the collisions of ballistic particles interacting with an irregular surface modeled by a quadratic Koch curve, are studied numerically. The q moments of the source spatial distribution of collision numbers μ(x) are characterized by a sequence of "collision exponent" τ(q). The measure μ(x) is found to be multifractal even when a random micro-roughness (or random re-emission) of the surface exists. The dimensions f(α), obtained by a Legendre transfo

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Pereira de Siqueira Campos, Antonio Luiz, Robson Hebraico Cipriano Maniçoba, and Adaildo Gomes d’Assunção. "Investigation of Enhancement Band Using Double Screen Frequency Selective Surfaces with Koch Fractal Geometry at Millimeter Wave Range." Journal of Infrared, Millimeter, and Terahertz Waves 31, no. 12 (2010): 1503–11. http://dx.doi.org/10.1007/s10762-010-9735-8.

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Macdonald, Blair. "Experiment on Inverted Fractal Corresponds with Cosmological Observations and Conjectures." International Journal of Quantum Foundations 6, no. 1 (2023): 1–32. https://doi.org/10.5281/zenodo.10493227.

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In a recent paper—The Fractal Corresponds to Light and Quantum Foundation Problems—the author described the forward-looking progressive perspective of a fractal. In this paper the author modelled the complementary and dual perspective of the same fractal: its retrospective perspective, this time it pertaining specifically to cosmology. Recent discussions as to whether the universe is fractal concluded with the 2012 WiggleZ Dark Energy Survey findings in (part) agreement with fractal-cosmology proponents that the small-scale observable universe is fractal; however, beyond this, the

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Huang, Yifan, Chao Liu, Bin Yao, Yan Yin, and Lei Bi. "Scattering matrices of mineral dust aerosols: a refinement of the refractive index impact." Atmospheric Chemistry and Physics 20, no. 5 (2020): 2865–76. http://dx.doi.org/10.5194/acp-20-2865-2020.

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Abstract. Mineral dust, as one of the most important aerosols, plays a crucial role in the atmosphere by directly interacting with radiation, while there are significant uncertainties in determining dust optical properties to quantify radiative effects and to retrieve their properties. Laboratory and in situ measurements of the refractive indices (RIs) of dust differ, and different RIs have been applied in numerical studies used for model developments, aerosol retrievals, and radiative forcing simulations. This study reveals the importance of the dust RI for the development of a model of dust

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ZANTEMA, HANS. "TURTLE GRAPHICS OF MORPHIC SEQUENCES." Fractals 24, no. 01 (2016): 1650009. http://dx.doi.org/10.1142/s0218348x16500092.

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The simplest infinite sequences that are not ultimately periodic are pure morphic sequences: fixed points of particular morphisms mapping single symbols to strings of symbols. A basic way to visualize a sequence is by a turtle curve: for every alphabet symbol fix an angle, and then consecutively for all sequence elements draw a unit segment and turn the drawing direction by the corresponding angle. This paper investigates turtle curves of pure morphic sequences. In particular, criteria are given for turtle curves being finite (consisting of finitely many segments), and for being fractal or sel

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Yassen, M. T., H. A. Hammas, M. R. Hussan, A. J. Salim, and J. K. Ali. "Design of Compact Dual-band Fractal Monopole Antenna with Virtually Extended Ground Plane." Advanced Electromagnetics 7, no. 4 (2018): 19–26. http://dx.doi.org/10.7716/aem.v7i4.725.

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Achieving a particular response to serve multiple wireless applications is regarded as the primary demand in our modern age because of the considerable development of the communication devices. In this paper, a compact monopole antenna with reduced ground plane has been suggested to meet the requirements of the dual-band WLAN applications. The antenna miniaturization has been carried out in employing two techniques. Initially, the fractal geometry has been applied to the antenna radiating element. Two-sided Koch fractal curves up to the third iteration have been used to increase the path of el

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Buriboev, Abror Shavkatovich, Djamshid Sultanov, Zulaykho Ibrohimova, and Heung Seok Jeon. "Mathematical Modeling and Recursive Algorithms for Constructing Complex Fractal Patterns." Mathematics 13, no. 4 (2025): 646. https://doi.org/10.3390/math13040646.

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In this paper, we present mathematical geometric models and recursive algorithms to generate and design complex patterns using fractal structures. By applying analytical, iterative methods, iterative function systems (IFS), and L-systems to create geometric models of complicated fractals, we developed fractal construction models, visualization tools, and fractal measurement approaches. We introduced a novel recursive fractal modeling (RFM) method designed to generate intricate fractal patterns with enhanced control over symmetry, scaling, and self-similarity. The RFM method builds upon traditi

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Жихарев and L. Zhikharev. "Generalization to Three-Dimensional Space Fractals of Pythagoras and Koch. Part I." Geometry & Graphics 3, no. 3 (2015): 24–37. http://dx.doi.org/10.12737/14417.

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Fractals are geometric objects, each part of which is similar to the whole object, so that if we take a part and increase its size to the size of the whole object, it would be impossible to notice a difference. In other words, fractals are sets having scale invariance. In mathematics, they are associated primarily with non-differentiable functions. The concept of &#34;fractal&#34; (from the Latin &#34;Fractus&#34; meaning «broken») had been introduced by Benoit Mandelbrot (1924–2010), French and American mathematician, physicist,

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Zhao, Pengcheng, Kai Zhang, and Zichen Deng. "Elastic Wave Propagation in Lattice Metamaterials with Koch Fractal." Acta Mechanica Solida Sinica 33, no. 5 (2020): 600–611. http://dx.doi.org/10.1007/s10338-020-00177-w.

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Abstract 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 respo

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Zhikharev, L. "Application of the Koch Curve to Increase the Strength of Aircraft Parts." Geometry & Graphics 10, no. 4 (2023): 13–25. http://dx.doi.org/10.12737/2308-4898-2022-10-4-13-25.

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Fractals are formed by iterative repetition of the construction algorithm at different scale levels. The use of such an algorithm, which increases the strength properties during the construction of the structure, will strengthen these properties with each iteration. The Koch curve principle is applied in the article. Replacing the compressible plate with four new ones connected at angles increases the stability of the structure. This article theoretically confirms the increase in the stability of the Koch plate both at the level of individual plates and at the level of fractal segments a

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