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Journal articles on the topic 'Fractal mathematics'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Alghar, Muhammad Zia, Natasya Ziana Walidah, and Marhayati Marhayati. "Ethnomathematics: The exploration of fractal geometry in Tian Ti Pagoda using the Lindenmayer system." Alifmatika: Jurnal Pendidikan dan Pembelajaran Matematika 5, no. 1 (2023): 57–69. http://dx.doi.org/10.35316/alifmatika.2023.v5i1.57-69.

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This study explores the concept of fractal geometry found in the Tian Ti Pagoda. Fractal geometry is a branch of mathematics describing the properties and shapes of various fractals. A qualitative method with an ethnographic approach is used in this study. Observation, field notes, interviews, documentation, and literature study obtained research data. The observation results were processed computationally using the Lindenmayer system method via the L-Studio application to view fractal shapes. The results show that the concept of fractal geometry is contained in the ornaments on the Tian Ti Pa

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Siegrist, Raymond. "Activities for Students: Inquiry into Fractals." Mathematics Teacher 103, no. 3 (2009): 206–12. http://dx.doi.org/10.5951/mt.103.3.0206.

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The exotic images of fractals often pique the interest of high school mathematics students, and this interest presents an opportunity for geometry teachers to draw students into an investigation of transformations and patterns. By using a simple building block and fractals' self-imaging characteristic (as the figure grows, it retains the pattern established by the building block), teachers can bring construction of fractals into the high school geometry curriculum. The three activities described in this article engage students in constructing a fractal, searching a fractal for patterns, and us

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Siegrist, Raymond. "Activities for Students: Inquiry into Fractals." Mathematics Teacher 103, no. 3 (2009): 206–12. http://dx.doi.org/10.5951/mt.103.3.0206.

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The exotic images of fractals often pique the interest of high school mathematics students, and this interest presents an opportunity for geometry teachers to draw students into an investigation of transformations and patterns. By using a simple building block and fractals' self-imaging characteristic (as the figure grows, it retains the pattern established by the building block), teachers can bring construction of fractals into the high school geometry curriculum. The three activities described in this article engage students in constructing a fractal, searching a fractal for patterns, and us

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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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MITINA, OLGA V., and FREDERICK DAVID ABRAHAM. "THE USE OF FRACTALS FOR THE STUDY OF THE PSYCHOLOGY OF PERCEPTION: PSYCHOPHYSICS AND PERSONALITY FACTORS, A BRIEF REPORT." International Journal of Modern Physics C 14, no. 08 (2003): 1047–60. http://dx.doi.org/10.1142/s0129183103005182.

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The present article deals with perception of time (subjective assessment of temporal intervals), complexity and aesthetic attractiveness of visual objects. The experimental research for construction of functional relations between objective parameters of fractals' complexity (fractal dimension and Lyapunov exponent) and subjective perception of their complexity was conducted. As stimulus material we used the program based on Sprott's algorithms for the generation of fractals and the calculation of their mathematical characteristics. For the research 20 fractals were selected which had differen

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TREGUBOVA, I. A. "FRACTAL GRAPHICS FOR VIRTUAL ENVIRONMENT GENERATION." Digital Technologies 26 (2019): 29–35. http://dx.doi.org/10.33243/2313-7010-26-29-35.

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Progress in hardware and software development is impressively fast. The main reason of computer graphics fast improvement is a full experience that can be reached though visual representation of our world. Therefore, the most interesting problem of it is a realistic image with high quality and resolution, which often requires procedural graphics generation. The article analyzes simplicity of a fractal and mathematics abstraction. Mathematics describes not only accuracy and logic but also beauty of the Universe. Mountains, clouds, trees, cells do not fit into the world of Euclidean geometry. Th

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Жихарев, Л., and L. Zhikharev. "Fractals In Three-Dimensional Space. I-Fractals." Geometry & Graphics 5, no. 3 (2017): 51–66. http://dx.doi.org/10.12737/article_59bfa55ec01b38.55497926.

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It has long been known that there are fractals, which construction resolve into cutting out of elements from lines, curves or geometric shapes according to a certain law. If the fractal is completely self-similar, its dimensionality is reduced relative to the original object and usually becomes fractional. The whole fractal is often decomposing into a set of separate elements, organized in the space of corresponding dimension. German mathematician Georg Cantor was among the first to propose such fractal set in the late 19th century. Later in the early 20th century polish mathematician Vaclav S

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Sergeeva, Larisa. "THE USE OF FRACTAL GEOMETRY ELEMENTS AS A MEANS OF AESTHETIC EDUCATION OF PRIMARY SCHOOL STUDENTS." SOCIETY. INTEGRATION. EDUCATION. Proceedings of the International Scientific Conference 3 (May 20, 2020): 562. http://dx.doi.org/10.17770/sie2020vol3.4934.

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Currently, the issue of aesthetic education is becoming one of the key issues in the field of education. The relevance of the article is due to the fact that mathematics has great aesthetic potential, which is not always revealed in the process of teaching mathematics in elementary school. An important role in revealing the beauty of mathematical content in elementary school is played by the familiarity of schoolchildren with a young rapidly developing mathematical field - fractal geometry. In order to prove the effectiveness of using fractal geometry elements for the formation of aesthetic re

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LAI, PENG-JEN. "HOW TO MAKE FRACTAL TILINGS AND FRACTAL REPTILES." Fractals 17, no. 04 (2009): 493–504. http://dx.doi.org/10.1142/s0218348x09004533.

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Intensive research on fractals began around 1980 and many new discoveries have been made. However, the connection between fractals, tilings and reptiles has not been thoroughly explored. This paper shows that a method, similar to that used to construct irregular tilings in ℜ2 can be employed to construct fractal tilings. Five main methods, including methods in Escher style paintings and the Conway criterion are used to create the fractal tilings. Also an algorithm is presented to generate fractal reptiles. These methods provide a more geometric way to understand fractal tilings and fractal rep

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LAPIDUS, MICHEL L. "FRACTALS AND VIBRATIONS: CAN YOU HEAR THE SHAPE OF A FRACTAL DRUM?" Fractals 03, no. 04 (1995): 725–36. http://dx.doi.org/10.1142/s0218348x95000643.

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We study various aspects of the question “Can one hear the shape of a fractal drum?”, both for “drums with fractal boundary” (or “surface fractals”) and for “drums with fractal membrane” (or “mass fractals”).

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Sulistyawati, Eka, and Imam Rofiki. "Ethnomathematics and creativity study in the construction of batik based on fractal geometry aided by GeoGebra." International Journal on Teaching and Learning Mathematics 4, no. 1 (2022): 15–28. http://dx.doi.org/10.18860/ijtlm.v5i1.10883.

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This study aims to describe geometric objects that used by students on constructing fractal batik using Geogebra, procedure that used to construct fractal batik design, and students creativity on the process of constructing fractal batik. The qualitative descriptive research was applies including data collection, data separation, data analysis and conclusions. The research data were obtained from 97 students of tadris mathematics IAIN Kediri. The research results showed that fractal batik was constructed from a single geometric shape and combination of 2, 3, and 4 single geometric shapes throu

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Karakus, Fatih. "A Cross-age study of students' understanding of fractals." Bolema: Boletim de Educação Matemática 27, no. 47 (2013): 829–46. http://dx.doi.org/10.1590/s0103-636x2013000400007.

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The purpose of this study is to examine how students understand fractals depending on age. Students' understandings were examined in four dimensions: defining fractals, determining fractals, finding fractal patterns rules and mathematical operations with fractals. The study was conducted with 187 students (grades 8, 9, 10) by using a two-tier test consisting of nine questions prepared based on the literature and Turkish mathematics and geometry curriculums. The findings showed that in all grades, students may have misunderstandings and lack of knowledge about fractals. Moreover, students can i

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Rezende, Veridiana, Mariana Moran, Thais Michele Mártires, and Fabricia Carvalho Paixão. "O Fractal Árvore Pitagórica e Diferentes Representações: uma Investigação com Alunos do Ensino Médio." Jornal Internacional de Estudos em Educação Matemática 11, no. 2 (2018): 160. http://dx.doi.org/10.17921/2176-5634.2018v11n2p160-171.

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A Geometria dos Fractais em sala de aula propicia diversas possibilidades de estudos de conceitos matemáticos, incentiva o uso de recursos tecnológicos, proporciona surpresas pela beleza e complexidade dos fractais. Apresentamos neste texto uma investigação acerca das possibilidades do uso de diferentes registros de representação semiótica aliado à Geometria dos Fractais, no que concerne ao ensino de matemática. E, para exemplificar estas possibilidades em sala de aula, relatamos uma situação de ensino, que foi implementada com quinze (15) alunos do 3º ano do Ensino Médio de uma escola pública

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Kern, Jane F., and Cherry C. Mauk. "Exploring Fractals a Problem—solving Adventure Using Mathematics and Logo." Mathematics Teacher 83, no. 3 (1990): 179–224. http://dx.doi.org/10.5951/mt.83.3.0179.

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The concept of fractal is relatively new yet is quickly being recognized as a realistic way to model nature. Fractals can be used to generate stimulating problems that can be solved using no more than high school algebra and geometry.

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Retnaningsih, Retnaningsih. "Fractal Geometry, Fibonacci Numbers, Golden Ratios, And Pascal Triangles as Designs." Journal of Academic Science 1, no. 1 (2024): 51–66. http://dx.doi.org/10.59613/msd4n328.

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Fractal geometry is a part of mathematics that discusses the shape of fractals or any form that is self-similarity. A fractal can be broken down into parts that are all similar to the original fractal. Fractals have infinite detail and can have self-similar structures at different magnifications. In many cases, a fractal can be generated by repeating a pattern, which is usually in a recursive or iterative process. In mathematics and art, two values are considered to be a golden ratio relationship if the ratio between the sum of the two values to the large value is equal to the ratio between th

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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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Adams, Thomasenia Lott, and Fatma Aslan-Tutak. "Math Roots: Serving Up Sierpinski!" Mathematics Teaching in the Middle School 11, no. 5 (2006): 248–53. http://dx.doi.org/10.5951/mtms.11.5.0248.

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The sierpinski triangle, created in 1916, has some very interesting characteristics. It is an impressive and valuable topic for mathematical exploration, since it combines Euclidean geometry (triangles and measurement) with fractal geometry. The Sierpinski triangle, also known as the Sierpinski gasket, is a fractal formed from an equilateral triangle. It is one of the most popular fractals to construct and analyze in middle school mathematics lessons. Since the 1960s, it has been possible to design fractals using a computer program, especially the complex examples that are often difficult to c

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Grinchenko, Victor, and Volodymyr Matsypura. "FRACTALIST." In the world of mathematics, no. 1 (2) (2024): 60–87. https://doi.org/10.17721/1029-4171.2024/2.9.

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The article is dedicated to the 100-th anniversary of the birth of Benoit Mandelbrot, the man who coined the word ''fractal'' and, in fact, became the author of a new direction in science - fractal geometry. The article briefly presents his biography, discusses the definition of the concept of fractal. Some attention is paid to Mandelbrot's predecessors and it is considered that it was B. Mandelbrot, thanks to his erudition and encyclopedic knowledge, who was able to systematize a huge number of previously known and newly discovered fractals. Today, fractals constitute an important section of

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SUZUKI, MASUO. "FRACTAL FORM ANALYSIS." Fractals 04, no. 03 (1996): 237–39. http://dx.doi.org/10.1142/s0218348x96000327.

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Mitic, Vojislav, Goran Lazovic, Jelena Manojlovic, et al. "Entropy and fractal nature." Thermal Science 24, no. 3 Part B (2020): 2203–12. http://dx.doi.org/10.2298/tsci191007451m.

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Existing, the biunivocal correspondents between the fractal nature and the nature discovered by fractals is the source and meeting point from those two aspects which are similar to the thermodynamically philosophical point of view. Sometimes we can begin from the end. We are substantial part of such fractals space nature. The mathematics fractal structures world have been inspired from nature and Euclidian geometry imagined shapes, and now it is coming back to nature serving it. All our analysis are based on several experimental results. The substance of the question regarding entropy and frac

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Buchanan, Mark. "Mathematics: A fractal life." Nature 490, no. 7421 (2012): 476–77. http://dx.doi.org/10.1038/490476a.

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Guerrero Torrenegra, Alejandro. "Periferias espontáneas de Ciudad Bolívar-Bogotá: vista desde la perspectiva fractal. Abordaje metodológico." Procesos Urbanos 3 (December 1, 2016): 89–98. http://dx.doi.org/10.21892/2422085x.269.

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Resumen: Las periferias internas espontáneas de América Latina se caracterizan por sus genes tipológicos informales-invasiones, que identifican a los tejidos informales que se producen como consecuencia de las construcciones y reconstrucciones auto-gestionadas. El objetivo principal de la investigación fue identificar la existencia de patrones o códigos fractales en las manzanas de la periferia interna espontánea de las UPZ, de la localidad de Ciudad Bolívar de Bogotá. La investigación es descriptiva y analítica, sigue pautas fenomenológicas, con matemática (no lineales) y geometría fractal. E

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Semkow, Thomas M. "Neighborhood Volume for Bounded, Locally Self-Similar Fractals." Fractals 05, no. 01 (1997): 23–33. http://dx.doi.org/10.1142/s0218348x97000048.

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We derive the formulas for neighborhood volume (Minkowski volume in d-dimensions) for fractals which have a curvature bias and are thus bounded. Both local surface fractal dimension and local mass fractal dimension are included as well as a radius of the neighborhood volume comparable with the size of the fractal. We consider two types of the neighborhood volumes: simplified and generalized, as well as the volumes below and above the fractal boundary. The formulas derived are generalizations of the equations for isotropic unbounded fractals. Based on the simplified-volume concept, we establish

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Denisova, L. V., and V. O. Dzhenzher. "AROUND THE SIERPINSKI TRIANGLE. THE CHAOS GAME AND THE SCRATCH PROGRAMMING." Informatics in school, no. 4 (July 3, 2020): 20–26. http://dx.doi.org/10.32517/2221-1993-2020-19-4-20-26.

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The article discusses the experience of studying Sierpinski geometric fractals in circle classes in mathematics and informatics with students of grades 4–6. A versatile approach to the construction of fractal objects is used: a method of cutting out of paper, Chaos Game on transparent films and implementing the studied algorithm in the Scratch programming language. At the same time, some issues related to the occurrence of random events are discussed with the children. After the fractal is obtained on a computer, the research phase of the constructed model follows. In the computer Chaos Game:

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Coppens, Marc-Olivier, and Gilbert F. Froment. "The Effectiveness of Mass Fractal Catalysts." Fractals 05, no. 03 (1997): 493–505. http://dx.doi.org/10.1142/s0218348x97000395.

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Many porous catalysts have a fractal surface, but only rarely do they have a fractal volume, the main exceptions being extremely porous aerogels. It has been suggested that a fractal shape of their volume would be ideal, because it has an infinite area per unit mass that is easily accessible by the reactants. This paper investigates the efficiency of mass fractals by comparing them with nonfractal catalysts. It is found that the specific surface areas of comparable nonfractal catalysts are of the same order of magnitude, if not higher than those of mass fractals. Despite the high effectiveness

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Ribeiro, P., and D. Queiros-Condé. "A scale-entropy diffusion equation to explore scale-dependent fractality." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2200 (2017): 20170054. http://dx.doi.org/10.1098/rspa.2017.0054.

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In the last three decades, fractal geometry became a mathematical tool widely used in physics. Nevertheless, it has been observed that real multi-scale phenomena display a departure to fractality that implies an impossibility to define the multi-scale features with an unique fractal dimension, leading to variations in the scale-space. The scale-entropy diffusion equation theorizes the organization of the scale dynamics involving scale-dependent fractals. A study of the theory is possible through the scale-entropy sink term in the equation and corresponds to precise behaviours in scale-space. I

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Perham, Arnold E., and Faustine L. Perham. "The Power of L-systems in Fractal Construction and Theory." Mathematics Teacher 98, no. 7 (2005): 459–67. http://dx.doi.org/10.5951/mt.98.7.0459.

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In searching the Web for an interactive applet that would help high school geometry students construct line fractals, we discovered a unique method called L–systems. Applets that integrate L–system theory with Turtle Geometry provide students with an easy–to–use graphics tool capable of creating a wide variety of fractals. In our experience, when students construct their own fractal drawings, rather than using prepared ones, they exhibit increased interest, insight, and enthusiasm. Mandelbrot and Frame (2002) make a strong case for fractals in the classroom, noting that the study of fractals h

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PE, JOSEPH L. "ANA'S GOLDEN FRACTAL." Fractals 11, no. 04 (2003): 309–13. http://dx.doi.org/10.1142/s0218348x03002269.

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In his fascinating book Wonders of Numbers, Clifford Pickover introduces the Ana sequence and fractal, two self-referential constructions arising from the use of language. This paper answers Pickover's questions on the relative composition of sequence terms and the dimension of the fractal. In the process, it introduces a novel way of obtaining fractals from iterative set operations. Also, it presents a beautiful variant of the Ana constructions involving the golden ratio. In conclusion, it suggests ways of constructing similar fractals for the Morse-Thue and "Look and Say" sequences.

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Shang, Yilun. "More on the normalized Laplacian Estrada index." Applicable Analysis and Discrete Mathematics 8, no. 2 (2014): 346–57. http://dx.doi.org/10.2298/aadm140724011s.

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Let G be a simple graph of order N. The normalized Laplacian Estrada index of G is defined as NEE(G)=?Ni=1 e?i?1, where ?1, ?2,... , ?N are the normalized Laplacian eigenvalues of G. In this paper, we give a tight lower bound for NEE of general graphs. We also calculate NEE for a class of treelike fractals, which contains T fractal and Peano basin fractal as its limiting cases. It is shown that NEE scales linearly with the order of the fractal, in line with a best possible lower bound for connected bipartite graphs.

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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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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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Babič, Matej, Jurij Mihelič, and Michele Calì. "Complex Network Characterization Using Graph Theory and Fractal Geometry: The Case Study of Lung Cancer DNA Sequences." Applied Sciences 10, no. 9 (2020): 3037. http://dx.doi.org/10.3390/app10093037.

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This paper discusses an approach developed for exploiting the local elementary movements of evolution to study complex networks in terms of shared common embedding and, consequently, shared fractal properties. This approach can be useful for the analysis of lung cancer DNA sequences and their properties by using the concepts of graph theory and fractal geometry. The proposed method advances a renewed consideration of network complexity both on local and global scales. Several researchers have illustrated the advantages of fractal mathematics, as well as its applicability to lung cancer researc

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JAMPOUR, MAHDI, MAHDI YAGHOOBI, MARYAM ASHOURZADEH, and ADEL SOLEIMANI. "A NEW FAST TECHNIQUE FOR FINGERPRINT IDENTIFICATION WITH FRACTAL AND CHAOS GAME THEORY." Fractals 18, no. 03 (2010): 293–300. http://dx.doi.org/10.1142/s0218348x10005020.

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Fingerprints are one of the simplest and most reliable human biometric features for identification. Geometry of the fingerprint is fractal and we can classify a fingerprint database with fractal dimension, but one can't identify a fingerprint with fractal dimension uniquely. In this paper we present a new approach for identifying fingerprint uniquely; for this purpose a new fractal is initially made from a fingerprint by using Fractal theory and Chaos Game theory. While making the new fractal, five parameters that can be used in identification process can be achieved. Finally a fractal is made

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Saeed, Mohammad. "Fractals Analysis of Cardiac Arrhythmias." Scientific World JOURNAL 5 (2005): 691–701. http://dx.doi.org/10.1100/tsw.2005.81.

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Heart rhythms are generated by complex self-regulating systems governed by the laws of chaos. Consequently, heart rhythms have fractal organization, characterized by self-similar dynamics with long-range order operating over multiple time scales. This allows for the self-organization and adaptability of heart rhythms under stress. Breakdown of this fractal organization into excessive order or uncorrelated randomness leads to a less-adaptable system, characteristic of aging and disease. With the tools of nonlinear dynamics, this fractal breakdown can be quantified with potential applications to

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MILLÁN, H., M. AGUILAR, J. DOMÌNGUEZ, L. CÈSPEDES, E. VELASCO, and M. GÒNZALEZ. "A NOTE ON THE PHYSICS OF SOIL WATER RETENTION THROUGH FRACTAL PARAMETERS." Fractals 14, no. 02 (2006): 143–48. http://dx.doi.org/10.1142/s0218348x06003131.

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Fractals are important for studying the physics of water transport in soils. Many authors have assumed a mass fractal structure while others consider a fractal surface approach. Each model needs comparisons on the same data set in terms of goodness-of-fit and physical interpretation of parameters. In this note, it is shown, with some representative data sets, that a pore-solid interface fractal model could fit soil water retention data better than a mass fractal model. In addition to the interfacial fractal dimension, this model predicts the tension at dryness. This value is very close to 106

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MEI, MAOFEI, BOMING YU, JIANCHAO CAI, and LIANG LUO. "A HIERARCHICAL MODEL FOR MULTI-PHASE FRACTAL MEDIA." Fractals 18, no. 01 (2010): 53–64. http://dx.doi.org/10.1142/s0218348x1000466x.

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The size distributions of solid particles and pores in porous media are approximately hierarchical and statistically fractals. In this paper, a model for single-phase fractal media is constructed, and the analytical expressions for area, fractal dimension and distribution function for solid particles are derived. The distribution function of solid particles obtained from the proposed model is in good agreement with available experimental data. Then, a model for approximate two-phase fractal media is developed. Good agreement is found between the predicted fractal dimensions for pore space from

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IGNATOWICH, MICHAEL J., DANIEL J. KELLEHER, CATHERINE E. MALONEY, DAVID J. MILLER, and KHRYSTYNA SERHIYENKO. "RESISTANCE SCALING FACTOR OF THE PILLOW AND FRACTALINA FRACTALS." Fractals 23, no. 02 (2015): 1550018. http://dx.doi.org/10.1142/s0218348x15500188.

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Much is known in the analysis of a finitely ramified self-similar fractal when the fractal has a harmonic structure: a Dirichlet form which respects the self-similarity of a fractal. What is still an open question is when such a structure exists in general. In this paper, we introduce two fractals, the fractalina and the pillow, and compute their resistance scaling factor. This is the factor which dictates how the Dirichlet form scales with the self-similarity of the fractal. By knowing this factor one can compute the harmonic structure on the fractal. The fractalina has scaling factor [Formul

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XIANG, ZHIYANG, KAI-QING ZHOU, and YIBO GUO. "GAUSSIAN MIXTURE NOISED RANDOM FRACTALS WITH ADVERSARIAL LEARNING FOR AUTOMATED CREATION OF VISUAL OBJECTS." Fractals 28, no. 04 (2020): 2050068. http://dx.doi.org/10.1142/s0218348x20500681.

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Because of the self-similarity properties of nature, fractals are widely adopted as generators of natural object multimedia contents. Unfortunately, fractals are difficult to control due to their iterated function systems, and traditional researches on fractal generating visual objects focus on mathematical manipulations. In Generative Adversarial Nets (GANs), visual object generators can be automatically guided by a single image. In this work, we explore the problem of guiding fractal generators with GAN. We assume that the same category of fractal patterns is produced by a group of parameter

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Grigorchuk, Rostislav, and Supun Samarakoon. "Integrable and Chaotic Systems Associated with Fractal Groups." Entropy 23, no. 2 (2021): 237. http://dx.doi.org/10.3390/e23020237.

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Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. T

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ZENG, QIUHUA, and HOUQIANG LI. "DIFFUSION EQUATION FOR DISORDERED FRACTAL MEDIA." Fractals 08, no. 01 (2000): 117–21. http://dx.doi.org/10.1142/s0218348x00000123.

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The movement of the fractal Brownian particle in isotropic and homogeneous two-dimensional assembling fractal spaces is studied by the standard diffusion equation on fractals, and we find that particle movement belongs to the anomalous diffusion. At the same time, by discussing the defectiveness of earlier proposed equations, a general form of analytic fractional diffusion equation is proposed for description of probability density of particles diffusing on fractal geometry at fractal time, and the solution connects with the ordinary solutions in the normal space time limit.

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STRUZIK, ZBIGNIEW R. "SOLVING THE TWO-DIMENSIONAL INVERSE FRACTAL PROBLEM WITH THE WAVELET TRANSFORM." Fractals 04, no. 04 (1996): 469–75. http://dx.doi.org/10.1142/s0218348x96000583.

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The methodology of the solution to the inverse fractal problem with the wavelet transform1,2 is extended to two-dimensional self-affine functions. Similar to the one-dimensional case, the two-dimensional wavelet maxima bifurcation representation used is derived from the continuous wavelet decomposition. It possesses translational and scale invariance necessary to reveal the invariance of the self-affine fractal. As many fractals are naturally defined on two-dimensions, this extension constitutes an important step towards solving the related inverse fractal problem for a variety of fractal type

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Gluchoff, Alan. "Hands-On Fractals and the Unexpected in Mathematics." Mathematics Teacher 99, no. 8 (2006): 570–75. http://dx.doi.org/10.5951/mt.99.8.0570.

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Within the subject of fractal geometry exist many examples of what we might call the unexpected in mathematics. Although there are many articles on fractal–building projects, one aspect of this material that provides such an example, the Contraction Mapping theorem, appears to be underrepresented. The purpose of this article is to provide a hands–on activity that illustrates this topic. We produce fractal images by iterating a set of simple rules to yield a sequence of images that in the limit consists of a fractal form. In the process, students are confronted with a surprising outcome that th

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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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Hewelt, Blake, Haiqing Li, Mohit Kumar Jolly, Prakash Kulkarni, Isa Mambetsariev, and Ravi Salgia. "The DNA walk and its demonstration of deterministic chaos—relevance to genomic alterations in lung cancer." Bioinformatics 35, no. 16 (2019): 2738–48. http://dx.doi.org/10.1093/bioinformatics/bty1021.

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AbstractMotivationAdvancements in cancer genetics have facilitated the development of therapies with actionable mutations. Although mutated genes have been studied extensively, their chaotic behavior has not been appreciated. Thus, in contrast to naïve DNA, mutated DNA sequences can display characteristics of unpredictability and sensitivity to the initial conditions that may be dictated by the environment, expression patterns and presence of other genomic alterations. Employing a DNA walk as a form of 2D analysis of the nucleotide sequence, we demonstrate that chaotic behavior in the sequence

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JÁNOSI, IMRE M., and ANDRÁS CZIRÓK. "FRACTAL CLUSTERS AND SELF-ORGANIZED CRITICALITY." Fractals 02, no. 01 (1994): 153–68. http://dx.doi.org/10.1142/s0218348x94000156.

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Self-organized criticality (SOC) and fractals have been shown to be related in various ways. On the one hand, the original idea of SOC suggests that the common explanation of the origin of fractal shapes in nature may be based on self-organized processes. Thus different models exhibiting SOC result in relaxation clusters or avalanches whose geometrical characteristics could be described by fractals. On the other hand, there exist several models for fractal growth phenomena, such as viscous fingering, invasion percolation, dielectric breakdown, etc., and it is possible that the concept of SOC m

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ZHANG, XIN-MIN, L. RICHARD HITT, BIN WANG, and JIU DING. "SIERPIŃSKI PEDAL TRIANGLES." Fractals 16, no. 02 (2008): 141–50. http://dx.doi.org/10.1142/s0218348x08003934.

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We generalize the construction of the ordinary Sierpiński triangle to obtain a two-parameter family of fractals we call Sierpiński pedal triangles. These fractals are obtained from a given triangle by recursively deleting the associated pedal triangles in a manner analogous to the construction of the ordinary Sierpiński triangle, but their fractal dimensions depend on the choice of the initial triangles. In this paper, we discuss the fractal dimensions of the Sierpiński pedal triangles and the related area ratio problem, and provide some computer-generated graphs of the fractals.

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Behnamian, Sara, Saman Behnamian, Fatemeh Fogh, Firooz Pashaei, and Malihe Mahin Saran. "NOVELTY ARCHITECTURE AND MATHEMATICS IN AN IRANIAN MOSQUE." Journal of Islamic Architecture 6, no. 1 (2020): 7–12. http://dx.doi.org/10.18860/jia.v6i1.5508.

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Islamic architecture, particularly mosques architecture, has mainly been the focus of many architectural exhibitions in Muslim-majority countries. Recently, it has been influenced by novelty architecture and has been evolved into elaborate structures. Quds mosque in Tehran, Iran, is a picturesque architecture feat of a modern outlook that came under a lot of criticism for abandoning the traditional symbols of Islamic architecture. This study observes the Quds mosque from a mathematical standpoint using fractals as the method. Fractals are geometric constructions that exhibit similar or identic

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BAK, PER, and MAYA PACZUSKI. "THE DYNAMICS OF FRACTALS." Fractals 03, no. 03 (1995): 415–29. http://dx.doi.org/10.1142/s0218348x95000345.

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Fractals are formed by avalanches, driving the system toward a critical state. This critical state is a fractal in d spatial plus one temporal dimension. Long range spatial and temporal properties are described by different cuts in this fractal attractor. We unify the origin of fractals, 1/f noise, Hurst exponents, Levy flights, and punctuated equilibria in terms of avalanche dynamics, and elucidate their relationships through analytical and numerical studies of simple models.

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CAI, JIANCHAO, FERNANDO SAN JOSÉ MARTÍNEZ, MIGUEL ANGEL MARTÍN, and EDMUND PERFECT. "AN INTRODUCTION TO FLOW AND TRANSPORT IN FRACTAL MODELS OF POROUS MEDIA: PART I." Fractals 22, no. 03 (2014): 1402001. http://dx.doi.org/10.1142/s0218348x14020010.

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This special issue gathers together a number of recent papers on fractal geometry and its applications to the modeling of flow and transport in porous media. The aim is to provide a systematic approach for analyzing the statics and dynamics of fluids in fractal porous media by means of theory, modeling and experimentation. The topics covered include lacunarity analyses of multifractal and natural grayscale patterns, random packing's of self-similar pore/particle size distributions, Darcian and non-Darcian hydraulic flows, diffusion within fractals, models for the permeability and thermal condu

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Cristea, Ligia L., and Bertran Steinsky. "Curves of infinite length in labyrinth fractals." Proceedings of the Edinburgh Mathematical Society 54, no. 2 (2011): 329–44. http://dx.doi.org/10.1017/s0013091509000169.

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AbstractWe define an infinite class of fractals, called horizontally and vertically blocked labyrinth fractals, which are dendrites and special Sierpiński carpets. Between any two points in the fractal there is a unique arc α; the length of α is infinite and the set of points where no tangent to α exists is dense in α.

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