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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 (October 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 different fractal dimensions that varied from 0.52 to 2.36, and the Lyapunov exponent from 0.01 to 0.22. We conducted two experiments: (1) A total of 20 fractals were shown to 93 participants. The fractals were displayed on the screen of a computer for randomly chosen time intervals ranging from 5 to 20 s. For each fractal displayed, the participant responded with a rating of the complexity and attractiveness of the fractal using ten-point scale with an estimate of the duration of the presentation of the stimulus. Each participant also answered the questions of some personality tests (Cattell and others). The main purpose of this experiment was the analysis of the correlation between personal characteristics and subjective perception of complexity, attractiveness, and duration of fractal's presentation. (2) The same 20 fractals were shown to 47 participants as they were forming on the screen of the computer for a fixed interval. Participants also estimated subjective complexity and attractiveness of fractals. The hypothesis on the applicability of the Weber–Fechner law for the perception of time, complexity and subjective attractiveness was confirmed for measures of dynamical properties of fractal images.
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Жихарев, Л., and L. Zhikharev. "Fractals In Three-Dimensional Space. I-Fractals." Geometry & Graphics 5, no. 3 (September 28, 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 Sierpinski described the Sierpinski carpet – one of the variants for the Cantor set generalization onto a two-dimensional space. At a later date the Austrian Karl Menger created a three-dimensional analogue of the Sierpinski fractal. Similar sets differ in a number of parameters from other fractals, and therefore must be considered separately. In this paper it has been proposed to call these fractals as i-fractals (from the Latin interfican – cut). The emphasis is on the three-dimensional i-fractals, created based on the Cantor and Sierpinski principles and other fractal dependencies. Mathematics of spatial fractal sets is very difficult to understand, therefore, were used computer models developed in the three-dimensional modeling software SolidWorks and COMPASS, the obtained data were processing using mathematical programs. Using fractal principles it is possible to create a large number of i-fractals’ three dimensional models therefore important research objectives include such objects’ classification development. In addition, were analyzed i-fractals’ geometry features, and proposed general principles for their creation.
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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 (July 2, 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 futuristic applications of fractals, whereas in the first part, we discussed the basics of fractals, mathematical description, fractal dimension and artistic applications. Among many different applications of fractals, fractal landscape generation (fractal landscapes that can simulate and describe natural terrains and landscapes more precisely by mathematical models of fractal geometry), fractal antennas (fractal-shaped antennas that are designed and used in devices which operate on multiple and wider frequency bands) and fractal image compression (a fractal-based lossy compression method for digital and natural images which uses inherent self-similarity present in an image) are the most creative, engineering-driven, industry-oriented, commercial and emerging applications. We consider each of these applications in detail along with some innovative and future ready applications.
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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 (November 30, 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 "fractal" (from the
 Latin "Fractus" meaning «broken») had been introduced by Benoit
 Mandelbrot (1924–2010), French and American mathematician,
 physicist, and economist. Mandelbrot had found that seemingly
 arbitrary fluctuations in price of goods have a certain tendency to
 change: it turned out that daily fluctuations are symmetrical with
 long-term price fluctuations. In fact, Benoit Mandelbrot applied
 his recursive (fractal) method to solve the problem. Since the last quarter of the nineteenth century, a large number
 of fractal curves and flat objects have been created; and methods
 for their application have been developed. From geometrical point
 of view, the most interesting fractals are "Koch snowflake" and
 "Pythagoras Tree". Two classes of analogues of the volumetric
 fractals were created with modern three-dimensional modeling
 program: "Fractals of growth” – like Pythagoras Tree, “Fractals of
 separation” – like Koch snowflake; the primary classification was
 developed, their properties were studied. Empiric data was processed
 with basic arithmetic calculations as well as with computer software.
 Among other things, for fractals of separation the task was to create
 an object with an infinite surface area, which in the future might
 acquire great importance for the development of the chemical and
 other industries.
5
Fraboni, Michael, and Trisha Moller. "Fractals in the Classroom." Mathematics Teacher 102, no. 3 (October 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 Euclidean objects? Fractals offer much to explore for even very young students.
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Fraboni, Michael, and Trisha Moller. "Fractals in the Classroom." Mathematics Teacher 102, no. 3 (October 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 Euclidean objects? Fractals offer much to explore for even very young students.
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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 (June 1, 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 larger than their size for each composing mass fractal. The power-law decay of the scattering intensityI(q) ∝ q^{D_{\rm s}-6}, where 2 <Ds< 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝r−τdr, withDs= τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface fractals, such as the fractal iteration number and the scaling factor.
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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 (August 1, 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 (November 12, 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 to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.
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Chen, Yanguang. "Fractal Modeling and Fractal Dimension Description of Urban Morphology." Entropy 22, no. 9 (August 30, 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 logic deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. Firstly, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Secondly, the topological dimension of city fractals based on the urban area is 0; thus, the minimum fractal dimension value of fractal cities is equal to or greater than 0. Thirdly, the fractal dimension of urban form is used to substitute the urban area, and it is better to define city fractals in a two-dimensional embedding space; thus, the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology.
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Siegrist, Raymond. "Activities for Students: Inquiry into Fractals." Mathematics Teacher 103, no. 3 (October 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 using transformations to build different fractals. Students gain insight into patterns as their fractals grow; they flip or rotate fractal pieces by following a design and translating the pieces into place.
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Siegrist, Raymond. "Activities for Students: Inquiry into Fractals." Mathematics Teacher 103, no. 3 (October 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 using transformations to build different fractals. Students gain insight into patterns as their fractals grow; they flip or rotate fractal pieces by following a design and translating the pieces into place.
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Pothiyodath, Nishanth, and Udayanandan Kandoth Murkoth. "Fractals and music." Momentum: Physics Education Journal 6, no. 2 (June 17, 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 dimension measurement. In this paper, we are suggesting that music which has an inherent fractal nature can be used as an object in classrooms to measure fractal dimensions. To find the fractal dimension we used the box-counting method. We studied the sound produced by different stringed instruments and some common noises. For good musical sound, the fractal dimension obtained is around 1.6882.
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LAPIDUS, MICHEL L. "FRACTALS AND VIBRATIONS: CAN YOU HEAR THE SHAPE OF A FRACTAL DRUM?" Fractals 03, no. 04 (December 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 (December 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 identify and determine the fractals, but when the grade level increased, this success decreases. Although students were able to intuitively determine a shape as fractal or not, they had some problems in finding pattern rules and formulizing them.
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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 (January 13, 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 formation of a fractal tree. These parameters affected the branching angle that can result in different fractal tree shape. Some random values of height and length parameters produced several variations of fractal tree. These values influenced the shape of fractal whether it tended to the left, to the right, or symmetrical shape.
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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 of DF is more accurately reconstructed with the S ePaC method in comparison with the BC method.
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Жихарев, Л., and L. Zhikharev. "Fractal Dimensionalities." Geometry & Graphics 6, no. 3 (November 14, 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 dimensionality should be less than the fractal’s topological dimension. Abram Samoilovich Bezikovich (1891–1970) was the author of first mathematical conclusions based on Felix Hausdorff (1868–1942) arguments and allowing determine the fractional dimensionality of multitudes. Bezikovich – Hausdorff dimensionality is determined through the multitude covering by unity elements. In practice, it is more convenient to use Minkowsky dimensionality for determining the fractional dimensionalities of fractals. There are also numerical methods for Minkowsky dimensionality calculation. In this study various approaches for fractional dimensionality determining are tested, dimensionalities of new fractals are defined. A broader view on the concept of dimensionality is proposed, its dependence on fractal parameters and interpretation of fractal sets’ structure are determined. An attempt for generalization of experimental dependences and determination of general regularities for fractals structure influence on their dimensionality is realized. For visualization of three-dimensional geometrical constructions, and plain evidence of empirical hypotheses were used computer models developed in the software for three-dimensional modeling (COMPASS, Inventor and SolidWorks), calculations were carried out in mathematical packages such as Wolfram Mathematica.
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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 (February 23, 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 – Box counting method and author's modification of Caliper method for fractal analysis of linear contours of anatomical structures. A fractal analysis of three linear fractals was performed: an artificial fractal – a Koch snowflake and two natural fractals – the outer contours of the pial surface of the human cerebellar vermis cortex and the cortex of the cerebral hemispheres. Fractal analysis was performed using the Box counting method and the author's modification of the Caliper method. The values of the fractal dimension of the artificial linear fractal (Koch snowflakes) obtained by the Caliper method coincide with the true value of the fractal dimension of this fractal, but the values of the fractal dimension obtained by the Box counting method do not match the true value of the fractal dimension. Therefore, fractal analysis of linear fractals using the Caliper method allows you to get more accurate results than the Box counting method. The values of the fractal dimension of artificial and natural fractals, calculated using the Box counting method, decrease with increasing image size and resolution; when using the Caliper method, fractal dimension values do not depend on these image parameters. The values of the fractal dimension of linear fractals, calculated using the Box counting method, increase with increasing width of the linear contour; the values calculated using the Caliper method do not depend on the contour line width. Thus, for the fractal analysis of linear fractals, preference should be given to the Caliper method and its modifications.
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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 (June 19, 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 status of people living in society.
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TARASOV, VASILY E. "ELECTROMAGNETIC FIELDS ON FRACTALS." Modern Physics Letters A 21, no. 20 (June 28, 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 (December 19, 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 fractals are demonstrated. In addition, the collage theorem on controlled Fisher fractals is established as well. The newly developing IFS and fractal set in the controlled metric space can provide the novel directions in the fractal theory.
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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 (October 13, 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 3D fractals. Fractal shapes can be applied for cultural development, namely to form wallpaper motifs. Fractal geometric motifs are obtained from iterating a function with the help of Wolframe Mathematica.
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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 (March 13, 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 study, I have considered this Pinwheel and its two intriguing Pinwheel properties of tiling fractals. These fractals have been considered as a graph and the same has been viewed under the scope of cordial and edge cordial labeling to apply this concept for further study in Engineering and science applications.</p>
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Semkow, Thomas M. "Neighborhood Volume for Bounded, Locally Self-Similar Fractals." Fractals 05, no. 01 (March 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 the procedure for calculating a distribution of physical quantities on bounded fractals and apply it to the distribution of trace elements in soil particles. Using the concept of the generalized volume, we show how an expectation value of a physical process can be calculated on bounded fractals, and apply it to the radon emanation from solid particles.
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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 (January 15, 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 generate an aesthetic experience and physiological stress reduction. I will discuss strategies for integrating fractals into building designs to induce positive impacts on the observer. Examples include fractal solar panels, fractal window shades, and fractal floor patterns. These applications of fractal fluency represent a fundamental and potentially impactful form of salutogenesis.
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PAVLOVITCH, BULAEV BORIS. "PHASE-PERIODIC STRUCTURES OF SELF-SIMILAR STAIRCASE FRACTALS." Fractals 08, no. 04 (December 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-periodical structure of the fractal. The staircase fractals are subject to the same rules that apply in information theory. It is shown that the difference in sharpness of holographic shapes is a direct consequence of spectra informations of the forms. The Hausdorff-Besicovitch dimension for this fractal has been estimated to be 0.5. On this view, the structure of staircase fractals is that of a Cantor Set. As indicated in the picture, such fractal resembles a "goose-wing." Also, the self-similar staircase fractal has much in common with a set of straight lines in a discrete coordinate system.
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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 point of view of theoretical analysis.
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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 (October 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 are checked by the inverse random midpoint displacement method (IRMD).
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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 (June 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 (August 2, 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 independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.
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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 (January 30, 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 random $V$-variable and homogeneous Markov constructions.
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LI, WEN XIA. "THE DIMENSION OF SETS DETERMINED BY THEIR CODE BEHAVIOR." Fractals 11, no. 04 (December 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) synthesized in labs and employed as gas sensors across materials, morphologies, and gas analytes. The focus is to investigate the morphology-driven gas response of these fab-fracs and identify key parameters of fractal geometry in influencing gas response. Fab-fracs with roughened microstructure, pore-network connectivity, and fractal dimension (D) less than 2 are projected to be possessing better gas sensing capabilities. Fab-fracs with these salient features will help in designing the commercial gas sensors with better performance.
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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 (January 1, 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 frontal lobes but also the highest beta in the parietal area, pointing to a complicated interplay between different parts of the brain when experiencing this pattern.
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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 (June 1, 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 highlight the basics of fractal theory and then show how fractals are useful in various geophysical studies by showing examples from potential fields and seismology and reservoir characterisation.
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BAK, PER, and MAYA PACZUSKI. "THE DYNAMICS OF FRACTALS." Fractals 03, no. 03 (September 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 (June 26, 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 different loss functions, opening up several potentials, e.g., learning fractals for downstream tasks, scientific understanding, etc.
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CHEN, YAN-GUANG. "FRACTAL TEXTURE AND STRUCTURE OF CENTRAL PLACE SYSTEMS." Fractals 28, no. 01 (February 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 central place models from the textural fractals. The method is theoretical deduction based on the dimension rules of fractal sets. The main results and findings are as follows. First, the central place fractals were formulated by the [Formula: see text] numbers and [Formula: see text] numbers. Second, three structural fractal models were constructed for central place systems according to the corresponding fractal dimensions. Third, the classic central place models proved to comprise Koch snowflake curve, Sierpinski space filling curve, and Gosper snowflake curve. Moreover, the traffic principle plays a leading role in urban and rural settlements evolution. A conclusion was reached that the textural fractal dimensions of central place models can be converted into the structural fractal dimensions and vice versa, and the structural dimensions can be directly used to appraise human settlement distributions in reality. Thus, the textural fractals can be indirectly employed to characterize the systems of human settlements.
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Maia Shevardenidze, Maia Shevardenidze. "Fractals and Forecasting of Economic Cycles." Economics 105, no. 09-10 (November 24, 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 tree branches, the fractal structure of the internal organs of a person, the psyche and organizational systems, etc. Recently, fractals have been widely used to study economic cycles. In particular, such a characteristic of a time series as fractal dimension makes it possible to determine the moment when the system becomes unstable and is ready to move to a new state. Modern science widely uses the theory of fractals to study time series in order to increase the reliability of forecasting economic dynamics. The main feature of a fractal is the infinite repetition of a self-similar structure in different scales. Analogous nature characterizes economic cycles. Samuelson and Hicks created an appropriate mathematical model for modeling economic cycles, later this approach was developed in the works of Goodwin. Recently, fractals have been widely used to study economic processes. To date, there are many different mathematical models of fractals, each of which differs from the others in a certain recursive function. With the help of methods of fractal analysis and their modifications, it is possible to determine moments of qualitative change in the behavior of the system and, accordingly, to predict their subsequent behavior in a relatively long-term perspective. The most urgent task, solved by the methods developed within this approach, is the identification of the moment of the onset of major economic crises, that is, forecasting them in the near future. Keywords: Fractals. Economic cycles. Prognosis.
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DEMÍR, BÜNYAMIN, ALI DENÍZ, ŞAHIN KOÇAK, and A. ERSIN ÜREYEN. "TUBE FORMULAS FOR GRAPH-DIRECTED FRACTALS." Fractals 18, no. 03 (September 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 (May 11, 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 response variations based on fractal geometry, and the prediction model provides acceptable errors, less than the natural variance of multiple experiments. Furthermore, the tree-like fractal samples showed an interesting behavior on the flexure test, where the fractals deformed uniformly and in a predictable pattern, enabling mechanical advantages in impact absorption applications.
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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 (June 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 parameters of initial patterns, affine transformations and random noises. Connections between these fractal parameters and visual objects are modeled by a Gaussian mixture model (GMM). Generator trainings are performed as gradients on GMM instead of fractals, so that evaluation numbers of iterated function systems are minimized. The proposed model requires no mathematical expertise from the user because parameters are trained by automatic procedures of GMM and GAN. Experiments include one 2D demonstration and three 3D real-world applications, where high-resolution visual objects are generated, and a user study shows the effectiveness of artificial intelligence guidances on fractals.
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COURTENS, E. "FRACTONS IN STRUCTURAL FRACTALS." Le Journal de Physique Colloques 24, no. C4 (April 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 (May 11, 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—the Sierpiński triangle (ST). To this aim, analytic expressions of small-angle scattering (SAS) intensities are derived, and it is shown that each type of deviation has its own unique imprint on the scattering curve. This allows the extraction of specific structural parameters, and thus the design and fabrication of artificial structures with pre-defined properties and functions. Moreover, the influence on the SAS intensity of various configurations induced in ST, can readily be extended to other 2D or 3D structures, allowing for exploration of structure-property relationships in various well-defined fractal geometries.
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Soltanifar, Mohsen. "A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals." Mathematics 9, no. 13 (July 1, 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 (April 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. In the first part of the paper, we study the scale space features when the scale-entropy sink term is modified. The second part is a numerical investigation and analysis of several solutions of the scale-entropy diffusion equation. By a precise measurement of the transition scales tested on truncated deterministic fractals, we developed a new simple method to estimate fractal dimension which appears to be much better than a classical method. Furthermore, we show that deterministic fractals display intrinsic log-periodic oscillations of the fractal dimension. In order to represent this complex behaviour, we introduce a departure to fractal diagram linking scale-space, scale-dependent fractal dimension and scale-entropy sink. Finally, we construct deterministic scale-dependent fractals and verify the results predicted by the scale-entropy diffusion equation.
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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 (June 27, 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 through steps (a) made basic patterns using geometric shapes in Geogebra, (b) Made New Tools to perform repetitions (iteration), (c) Determined the type of transformation that used to repeat the basic patterns, and (d) Constructed geometric fractal batik. Based on the creativity indicators fluency, 3 types of geometric fractals can be obtained, namely (a) Fractals with Repetition and Enlargement (FPB), (b) Fractals with Repetition and Change Position (FPS), and (c) Fractals with Mix Repetition (FPC). Based on the flexibility indicator, there are 32 basic geometric shapes that develop basic patterns by applying 15 types of transformation consist of Single Transformation, Double Transformation, Triple Transformation and Quadruple Transformation. Meanwhile, on the originality indicator, P3 is the basic shape that has been mostly developed into geometric fractal batik which is a combination of equilateral triangles and squares.
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Pulinchery, Prasanth, Nishanth Pothyiodath, and Udayanandan Kandoth Murkoth. "Chaos to fractals." Momentum: Physics Education Journal 7, no. 1 (January 10, 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.