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Journal articles on the topic 'Fractals'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

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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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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Husain, Akhlaq, Manikyala Navaneeth Nanda, Movva Sitaram Chowdary, and Mohammad Sajid. "Fractals: An Eclectic Survey, Part II." Fractal and Fractional 6, no. 7 (2022): 379. http://dx.doi.org/10.3390/fractalfract6070379.

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Fractals are geometric shapes and patterns that can describe the roughness (or irregularity) present in almost every object in nature. Many fractals may repeat their geometry at smaller or larger scales. This paper is the second (and last) part of a series of two papers dedicated to an eclectic survey of fractals describing the infinite complexity and amazing beauty of fractals from historical, theoretical, mathematical, aesthetical and technological aspects, including their diverse applications in various fields. In this article, our focus is on engineering, industrial, commercial and futuris

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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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Fraboni, Michael, and Trisha Moller. "Fractals in the Classroom." Mathematics Teacher 102, no. 3 (2008): 197–99. http://dx.doi.org/10.5951/mt.102.3.0197.

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What exactly is a fractal? Traditionally, students learn about the familiar forms of symmetry: reflection, translation, and rotation. Intuitively, fractals are symmetric with respect to magnification. A magnification of a small part of the fractal looks essentially the same as the entire picture. More formally, fractals have the property of self-similarity—that is, a fractal is any shape that is made up of smaller copies of itself. Self-similarity is what distinguishes fractals from most conventional Euclidean figures and makes them appealing. Do fractals hold the same characteristics as other

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Fraboni, Michael, and Trisha Moller. "Fractals in the Classroom." Mathematics Teacher 102, no. 3 (2008): 197–99. http://dx.doi.org/10.5951/mt.102.3.0197.

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What exactly is a fractal? Traditionally, students learn about the familiar forms of symmetry: reflection, translation, and rotation. Intuitively, fractals are symmetric with respect to magnification. A magnification of a small part of the fractal looks essentially the same as the entire picture. More formally, fractals have the property of self-similarity—that is, a fractal is any shape that is made up of smaller copies of itself. Self-similarity is what distinguishes fractals from most conventional Euclidean figures and makes them appealing. Do fractals hold the same characteristics as other

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Cherny, A. Yu, E. M. Anitas, V. A. Osipov, and A. I. Kuklin. "Scattering from surface fractals in terms of composing mass fractals." Journal of Applied Crystallography 50, no. 3 (2017): 919–31. http://dx.doi.org/10.1107/s1600576717005696.

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It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much

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Joy, Elizabeth K., and Dr Vikas Garg. "FRACTALS AND THEIR APPLICATIONS: A REVIEW." Journal of University of Shanghai for Science and Technology 23, no. 07 (2021): 1509–17. http://dx.doi.org/10.51201/jusst/21/07277.

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In this paper, I have discussed about fractals. The two key properties of fractals have been stated. A brief history about fractals is also mentioned. I have discussed about Mandelbrot fractal and have plotted it using python. A computer-generated fern is compared to a real fern to show how much fractals resemble the real-world objects. Various applications of fractal geometry have also been included.

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BANAKH, T., and N. NOVOSAD. "MICRO AND MACRO FRACTALS GENERATED BY MULTI-VALUED DYNAMICAL SYSTEMS." Fractals 22, no. 04 (2014): 1450012. http://dx.doi.org/10.1142/s0218348x14500121.

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Given a multi-valued function Φ : X ⊸ X on a topological space X we study the properties of its fixed fractal[Formula: see text], which is defined as the closure of the orbit Φω(*Φ) = ⋃n∈ωΦn(*Φ) of the set *Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals [Formula: see text] and [Formula: see text] for a contracting compact-valued function Φ : X ⊸ X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual t

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Chen, Yanguang. "Fractal Modeling and Fractal Dimension Description of Urban Morphology." Entropy 22, no. 9 (2020): 961. http://dx.doi.org/10.3390/e22090961.

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The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a scale-dependence measure, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for the scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, the question of how to understand city fractals is still pending. By means of lo

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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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Pothiyodath, Nishanth, and Udayanandan Kandoth Murkoth. "Fractals and music." Momentum: Physics Education Journal 6, no. 2 (2022): 119–28. http://dx.doi.org/10.21067/mpej.v6i2.6796.

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Many natural phenomena we find in our surroundings, are fractals. Studying and learning about fractals in classrooms is always a challenge for both teachers and students. We here show that the sound of musical instruments can be used as a good resource in the laboratory to study fractals. Measurement of fractal dimension which indicates how much fractal content is there, is always uncomfortable, because of the size of the objects like coastlines and mountains. A simple fractal source is always desirable in laboratories. Music serves to be a very simple and effective source for fractal dimensio

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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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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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Purnomo, Kosala Dwidja, Dita Wahyuningtyas, and Firdaus Ubaidillah. "Construction of Three Branches Fractal Trees Using Iterated Function System." Jurnal ILMU DASAR 23, no. 1 (2022): 9. http://dx.doi.org/10.19184/jid.v23i1.17447.

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There are two types of fractal: natural fractals and fractals set. The examples of natural fractals are trees, leaves, ferns, mountain, and coastlines. One of the examples of fractals set is Pythagorean tree. In the earlier study, the Pythagorean tree has two branches generated through several affine transformations, i.e dilation and rotation. Here, we developed the Pythagorean tree (or fractal tree) with three branches through dilation, translation, and rotation transformation using Iterated Function System (IFS) method. Some values of height and length parameters were selected to ensure the

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Dedovich, Tatiana, and Mikhail Tokarev. "Incomplete fractal showers and restoration of dimension." EPJ Web of Conferences 204 (2019): 06003. http://dx.doi.org/10.1051/epjconf/201920406003.

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The S ePaC and BC methods are used in the fractal analysis of mixed events containing incomplete fractals. The reconstruction of the event distribution by the dimension DF is studied. The procedures of analyzing incomplete fractals and correcting the determination of DF of combined fractals by the SePaC method are proposed. We find that the S ePaC method fully reconstructs incomplete fractals and suppresses background, the separation of the incomplete fractals and the background by the BC method depends on the basis of the formation of the fractal, and the distribution of events by the value o

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Жихарев, Л., and L. Zhikharev. "Fractal Dimensionalities." Geometry & Graphics 6, no. 3 (2018): 33–48. http://dx.doi.org/10.12737/article_5bc45918192362.77856682.

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One of the most important characteristics of a fractal is its dimensionality. In general, there are several options for mathematical definition of this value, but usually under the object dimensionality is understood the degree of space filling by it. It is necessary to distinguish the dimensionality of space and the dimension of multitude. Segment, square and cube are objects with dimensionality 1, 2 and 3, which can be in respective spaces: on a straight line, plane or in a 3D space. Fractals can have a fractional dimensionality. By definition, proposed by Bernois Mandelbrot, this fractional

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Maryenko, N. І., and O. Yu Stepanenko. "Fractal analysis of anatomical structures linear contours: modified Caliper method vs Box counting method." Reports of Morphology 28, no. 1 (2022): 17–26. http://dx.doi.org/10.31393/morphology-journal-2022-28(1)-03.

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Fractal analysis estimates the metric dimension and complexity of the spatial configuration of different anatomical structures. This allows the use of this mathematical method for morphometry in morphology and clinical medicine. Two methods of fractal analysis are most often used for fractal analysis of linear fractal objects: the Box counting method (Grid method) and the Caliper method (Richardson’s method, Perimeter stepping method, Ruler method, Divider dimension, Compass dimension, Yard stick method). The aim of the research is a comparative analysis of two methods of fractal analysis – Bo

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S, J. a. b. b. a. r. o. v. J. "FRACTAL STRUCTURE AND FRACTAL MEASUREMENT." 2022-yil, 3-son (133/1) ANIQ FANLAR SERIYASI 5, no. 129/2 (2021): 1–6. http://dx.doi.org/10.59251/2181-1296.v5.1292.855.

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. This article is devoted to the study of fractal structure and fractal dimensions. The article explains the concept of fractals, the ability to determine the fractal sizes of parts of the human body based on fractals and, based on these data, determine or predict what disease a person suffers from. In particular, the fractal structure of the human vascular system, the size of the fractal, etc. were calculated. In addition, fractals covered the causes of climate change, the causes of sudden waves in the seas and oceans, sudden changes in the economy, and even the improvement in the social stat

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TARASOV, VASILY E. "ELECTROMAGNETIC FIELDS ON FRACTALS." Modern Physics Letters A 21, no. 20 (2006): 1587–600. http://dx.doi.org/10.1142/s0217732306020974.

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Fractals are measurable metric sets with non-integer Hausdorff dimensions. If electric and magnetic fields are defined on fractal and do not exist outside of fractal in Euclidean space, then we can use the fractional generalization of the integral Maxwell equations. The fractional integrals are considered as approximations of integrals on fractals. We prove that fractal can be described as a specific medium.

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Thangaraj, C., and D. Easwaramoorthy. "Fractals via Controlled Fisher Iterated Function System." Fractal and Fractional 6, no. 12 (2022): 746. http://dx.doi.org/10.3390/fractalfract6120746.

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This paper explores the generalization of the fixed-point theorem for Fisher contraction on controlled metric space. The controlled metric space and Fisher contractions are playing a very crucial role in this research. The Fisher contraction on the controlled metric space is used in this paper to generate a new type of fractal set called controlled Fisher fractals (CF-Fractals) by constructing a system named the controlled Fisher iterated function system (CF-IFS). Furthermore, the interesting results and consequences of the controlled Fisher iterated function system and controlled Fisher fract

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Kediangan, Ilham Vanka Agustiawan, La Zakaria, and Agus Sutrisno. "Design 3D Wallpaper Motifs from Sierpinski Carpet Fractals Using Mathematica Applications." Journal of Mathematics: Theory and Applications 6, no. 2 (2024): 146–57. http://dx.doi.org/10.31605/jomta.v6i2.3569.

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The development of geometric studies is very rapid, including fractal geometry. A fractal is a geometric shape that fulfills the properties of the fractal dimension. If you look at a fractal at first glance, it has an irregular shape, but if you look further, there is a regularity. One of the properties of fractals is self-similarity which occurs when a fractal is enlarged. The fractal form has a pattern obtained from iterating a function with infinite repetition. Among the fractal shapes there is the Sierpinski carpet shape. Fractals in 2D form can be transformed using geometric concepts into

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A A, Sathakathulla. "Pinwheel tiling fractal graph- a notion to edge cordial and cordial labeling." International Journal of Applied Mathematical Research 5, no. 2 (2016): 84. http://dx.doi.org/10.14419/ijamr.v5i2.5700.

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<p>A fractal is a complex geometric figure that continues to display self-similarity when viewed on all scales. Tile substitution is the process of repeatedly subdividing shapes according to certain rules. These rules are also sometimes referred to as inflation and deflation rule. One notable example of a substitution tiling is the so-called Pinwheel tiling of the plane. Many examples of self-similar tiling are made of fractiles: tiles with fractal boundaries. . The pinwheel tiling was the first example of this sort. There are many as such as family of tiling fractal curves, but for my s

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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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Taylor, Richard. "The Potential of Biophilic Fractal Designs to Promote Health and Performance: A Review of Experiments and Applications." Sustainability 13, no. 2 (2021): 823. http://dx.doi.org/10.3390/su13020823.

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Fractal objects are prevalent in natural scenery. Their repetition of patterns at increasingly fine magnifications creates a rich complexity. Fractals displaying mid-range complexity are the most common and include trees, clouds, and mountains. The “fractal fluency” model states that human vision has adapted to process these mid-range fractals with ease. I will first discuss fractal fluency and demonstrate how it enhances the observer’s visual capabilities by focusing on experiments that have important practical consequences for improving the built environment. These enhanced capabilities gene

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PAVLOVITCH, BULAEV BORIS. "PHASE-PERIODIC STRUCTURES OF SELF-SIMILAR STAIRCASE FRACTALS." Fractals 08, no. 04 (2000): 323–35. http://dx.doi.org/10.1142/s0218348x00000378.

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The possibilities of investigating the self-similar staircase fractals in a discrete coordinate system are clearly very promising. Surprisingly, such fractals are found to have holographic properties. Some geometric shapes e.g. a circle or a quadrate, are produced by a well-defined boundary through generating the staircase fractals in 2D discrete space. The obtained "luminous boundaries" are remembered outer boundaries of any geometric form, regardless of the size. Actually, this process is similar to the photographing of an exterior form. Thus there is destruction of a natural phase-periodica

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Zheng, Hong Chan, Yi Li, Guo Hua Peng, and Ya Ning Tang. "A Multicontrol p-ary Subdivision Scheme to Generate Fractal Curves." Applied Mechanics and Materials 263-266 (December 2012): 1830–33. http://dx.doi.org/10.4028/www.scientific.net/amm.263-266.1830.

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In this paper, we firstly devise a p-ary subdivision scheme based on normal vectors with multi-parameters to generate fractals. The method is easy to use and effective in generating fractals since the values of the parameters and the directions of normal vectors can be designed freely to control the shape of generated fractals. Secondly, we illustrate the technique with some design results of fractal generation and the corresponding fractal examples from the point of view of visualization. Finally, some fractal properties of the limit of the presented subdivision scheme are described from the

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LIAW, SY-SANG, and FENG-YUAN CHIU. "CONSTRUCTING CROSSOVER-FRACTALS USING INTRINSIC MODE FUNCTIONS." Advances in Adaptive Data Analysis 02, no. 04 (2010): 509–20. http://dx.doi.org/10.1142/s1793536910000598.

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Real nonstationary time sequences are in general not monofractals. That is, they cannot be characterized by a single value of fractal dimension. It has been shown that many real-time sequences are crossover-fractals: sequences with two fractal dimensions — one for the short and the other for long ranges. Here, we use the empirical mode decomposition (EMD) to decompose monofractals into several intrinsic mode functions (IMFs) and then use partial sums of the IMFs decomposed from two monofractals to construct crossover-fractals. The scale-dependent fractal dimensions of these crossover-fractals

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Sander, Evelyn, Leonard M. Sander, and Robert M. Ziff. "Fractals and Fractal Correlations." Computers in Physics 8, no. 4 (1994): 420. http://dx.doi.org/10.1063/1.168501.

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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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Patiño Ortiz, Julián, Miguel Patiño Ortiz, Miguel-Ángel Martínez-Cruz, and Alexander S. Balankin. "A Brief Survey of Paradigmatic Fractals from a Topological Perspective." Fractal and Fractional 7, no. 8 (2023): 597. http://dx.doi.org/10.3390/fractalfract7080597.

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The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of indepe

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JÄRVENPÄÄ, ESA, MAARIT JÄRVENPÄÄ, ANTTI KÄENMÄKI, HENNA KOIVUSALO, ÖRJAN STENFLO, and VILLE SUOMALA. "Dimensions of random affine code tree fractals." Ergodic Theory and Dynamical Systems 34, no. 3 (2013): 854–75. http://dx.doi.org/10.1017/etds.2012.168.

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AbstractWe study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in

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LI, WEN XIA. "THE DIMENSION OF SETS DETERMINED BY THEIR CODE BEHAVIOR." Fractals 11, no. 04 (2003): 345–52. http://dx.doi.org/10.1142/s0218348x0300218x.

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By prescribing their code run behavior, we consider some subsets of Moran fractals. Fractal dimensions of these subsets are exactly obtained. Meanwhile, an interesting decomposition of Moran fractals is given.

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Kamathe, Vishal, and Rupali Nagar. "Morphology-driven gas sensing by fabricated fractals: A review." Beilstein Journal of Nanotechnology 12 (November 9, 2021): 1187–208. http://dx.doi.org/10.3762/bjnano.12.88.

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Fractals are intriguing structures that repeat themselves at various length scales. Interestingly, fractals can also be fabricated artificially in labs under controlled growth environments and be explored for various applications. Such fractals have a repeating unit that spans in length from nano- to millimeter range. Fractals thus can be regarded as connectors that structurally bridge the gap between the nano- and the macroscopic worlds and have a hybrid structure of pores and repeating units. This article presents a comprehensive review on inorganic fabricated fractals (fab-fracs) synthesize

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Hagerhall, Caroline M., Thorbjörn Laike, Richard P. Taylor, Marianne Küller, Rikard Küller, and Theodore P. Martin. "Investigations of Human EEG Response to Viewing Fractal Patterns." Perception 37, no. 10 (2008): 1488–94. http://dx.doi.org/10.1068/p5918.

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Owing to the prevalence of fractal patterns in natural scenery and their growing impact on cultures around the world, fractals constitute a common feature of our daily visual experiences, raising an important question: what responses do fractals induce in the observer? We monitored subjects' EEG while they were viewing fractals with different fractal dimensions, and the results show that significant effects could be found in the EEG even by employing relatively simple silhouette images. Patterns with a fractal dimension of 1.3 elicited the most interesting EEG, with the highest alpha in the fr

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Chauhan, Mahak Singh, Abhey Ram Bansal, and V. P. Dimri. "Scaling Laws and Fractal Geometry: Insights into Geophysical Data Interpretations." Journal Of The Geological Society Of India 101, no. 6 (2025): 983–89. https://doi.org/10.17491/jgsi/2025/174196.

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ABSTRACT Fractals, characterised by self-similarity and scale invariance, have emerged as powerful tools for understanding complex systems in geophysics. This paper highlights the applications of fractal geometry in geophysical data interpretation. For instance, fractal analysis is used in seismology to understand the fault systems, earthquake distribution, and the scaling laws governing seismic events. In potential fields, fractals are used to find the source depth, to design the optimum grid size of the survey, to detect the source and to separate signal from noise. In this paper, we first h

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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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Tu, Cheng-Hao, Hong-You Chen, David Carlyn, and Wei-Lun Chao. "Learning Fractals by Gradient Descent." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 2 (2023): 2456–64. http://dx.doi.org/10.1609/aaai.v37i2.25342.

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Fractals are geometric shapes that can display complex and self-similar patterns found in nature (e.g., clouds and plants). Recent works in visual recognition have leveraged this property to create random fractal images for model pre-training. In this paper, we study the inverse problem --- given a target image (not necessarily a fractal), we aim to generate a fractal image that looks like it. We propose a novel approach that learns the parameters underlying a fractal image via gradient descent. We show that our approach can find fractal parameters of high visual quality and be compatible with

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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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Maia Shevardenidze, Maia Shevardenidze. "Fractals and Forecasting of Economic Cycles." Economics 105, no. 09-10 (2022): 32–39. http://dx.doi.org/10.36962/ecs105/9-10/2022-32.

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For many centuries, scientists have used classical methods of calculation in the process of exploring and modeling the universe. And such functions that did not carry the proper smoothness or regularity were often considered as pathologies and were not given much attention. The fractal geometry of Benoit Mandelbrot deals with the study of such irregular sets. The basic concept of fractal geometry is a fractal, the origin of which is associated with computer modeling. With the help of fractals, it was possible to describe the glow of the sky during a thunderstorm, the dynamics of the growth of

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DEMÍR, BÜNYAMIN, ALI DENÍZ, ŞAHIN KOÇAK, and A. ERSIN ÜREYEN. "TUBE FORMULAS FOR GRAPH-DIRECTED FRACTALS." Fractals 18, no. 03 (2010): 349–61. http://dx.doi.org/10.1142/s0218348x10004919.

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Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.

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Stanciu Birlescu, Anca, Cristian Vilau, and Nicolae Balc. "Analysis of the Mechanical Behavior of Tree-like Fractal Structures in SLM-Manufactured Components." Materials 18, no. 10 (2025): 2215. https://doi.org/10.3390/ma18102215.

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Tree-like fractals as internal structures are a novel alternative to conventional lattice structures for mechanical components produced via Selective Laser Melting (SLM). This study explores the mechanical behavior of tree-like fractals, targeting flexure tests on SLM test samples manufactured using two distinct fractal configurations. The main objective is to develop numerical models that can predict the effect of the branching angle on the stress-strain curves, for both fractal configurations, from experimental flexure tests. A polynomial regression model is proposed to predict mechanical re

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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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COURTENS, E. "FRACTONS IN STRUCTURAL FRACTALS." Le Journal de Physique Colloques 24, no. C4 (1989): C4–143—C4–144. http://dx.doi.org/10.1051/jphyscol:1989422.

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Anitas, Eugen Mircea. "Structural Properties of Molecular Sierpiński Triangle Fractals." Nanomaterials 10, no. 5 (2020): 925. http://dx.doi.org/10.3390/nano10050925.

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The structure of fractals at nano and micro scales is decisive for their physical properties. Generally, statistically self-similar (random) fractals occur in natural systems, and exactly self-similar (deterministic) fractals are artificially created. However, the existing fabrication methods of deterministic fractals are seldom defect-free. Here, are investigated the effects of deviations from an ideal deterministic structure, including small random displacements and different shapes and sizes of the basic units composing the fractal, on the structural properties of a common molecular fractal

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Soltanifar, Mohsen. "A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals." Mathematics 9, no. 13 (2021): 1546. http://dx.doi.org/10.3390/math9131546.

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How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

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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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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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Pulinchery, Prasanth, Nishanth Pothyiodath, and Udayanandan Kandoth Murkoth. "Chaos to fractals." Momentum: Physics Education Journal 7, no. 1 (2023): 17–32. http://dx.doi.org/10.21067/mpej.v7i1.7502.

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In undergraduate classrooms, while teaching chaos and fractals, it is taught as if there is no relation between these two. By using some non linear oscillators we demonstrate that there is a connection between chaos and fractals. By plotting the phase space diagrams of four nonlinear oscillators and using box counting method of finding the fractal dimension we established the chaotic nature of the nonlinear oscillators. The awareness that all chaotic systems are good fractals will add more insights to the concept of chaotic systems.

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