Fractal Objects Research Articles

Application of Fractal Dimension for Cardiac Arrhythmias Classification

Fractal analysis is crucial for understanding complex, irregular patterns found in nature, finance, and various scientific fields. It helps to reveal self-similarity, where structures repeat at different scales, providing insights into chaotic systems like weather patterns, stock markets, and biological growth. By applying fractal analysis, researchers can model phenomena that traditional geometric methods cannot easily describe, enabling better predictions and deeper comprehension of dynamic systems. The Fractals are a fascinating mathematical tool for modeling the roughness of nature and understanding structure of such complex objects. They are considered a tool for understanding the world. In general, fractal objects are characterized by the fractal dimension. The application of fractal geometry to the analysis of ECG time series data is examined in this paper. A method based on the assessment of the Fractal Dimension (FD) of ECG recordings is suggested for the identification of cardiac diseases. In this work, and in order to exploit the fractal dimension to analyze fractal signals, the notion of fractal dimension is defined by presenting methods for calculating this dimension such as Higuchi algorithm, Katz method, regularization, box-counting etc… Each of them has its own advantages and disadvantages. This study has shown that the electrocardiogram (ECG) is a fractal signal. This allows to classify heartbeats founded on the concept of fractals. The main aim is to develop a digital technique to analyze ECG signals in order to make an accurate diagnosis of cardiovascular diseases.

Fractal Geometry, Fibonacci Numbers, Golden Ratios, And Pascal Triangles as Designs

Fractal geometry is a part of mathematics that discusses the shape of fractals or any form that is self-similarity. A fractal can be broken down into parts that are all similar to the original fractal. Fractals have infinite detail and can have self-similar structures at different magnifications. In many cases, a fractal can be generated by repeating a pattern, which is usually in a recursive or iterative process. In mathematics and art, two values are considered to be a golden ratio relationship if the ratio between the sum of the two values to the large value is equal to the ratio between the large value to the small value. A Fibonacci sequence is a sequence of numbers that has a unique shape. The first term of this sequence of numbers is 1, then the second term is also 1, then for the third term it is determined by adding the two previous terms so that a sequence of numbers with a certain pattern is obtained. Pascal's triangle is a geometric rule of binomial coefficients in a triangle. Shapes that resemble fractal geometry, golden ratios and Fibonacci numbers are found in many places in this realm, for example the shapes of various plants, animals, as well as in humans themselves, even the universe. This fractal geometry can also be used to design batik motif creations, architectural design or musical art. This paper describes how a fractal object can be made with geometric transformations that can be used to develop batik motif creations. And also look at the relationship between fractal geometry, the golden ratio, Fibonacci numbers, and Pascal's triangles and the shapes of these shapes that exist in this nature.

Continuity Dependence of Attractors of Iterated Function Systems with Orbital Condition

In the article Secelean and Wardowski (Qual Theory Dyn Syst 21:162, 2022), there were introduced iterated function systems involving certain condition of orbital type. The attractors of such IFSs substantially extend the family of fractal objects investigated so far. In the present work we provide some further properties and give the answer regarding the continuity dependence of this type of attractors with respect to the continuous change of the parameters that describe the maps of the given orbital IFS and the initially chosen compact subset.

From Zipf to Price and beyond

Consider a society comprising n members that are ranked in a decreasing order of their personal wealths. For this society: Zipf’s Law manifests the case in which the members’ ranks and wealths display an inverse power-law relation; and Price’s Law manifests the case in which the n richest members possess, collectively, half of all the wealth. This paper goes from Zipf’s Law to Price’s Law and beyond, and it does so by introducing and exploring a novel Generalized Price’s Law (GPL): a general allometric scaling of the quantiles of rank distributions. Akin to multifractals, the GPL is governed by spectrums of powers. The GPL spectra are investigated, and are shown to be fractal objects: from a socioeconomic perspective, the spectra are power-law Lorenz curves that characterize poor fractality and rich fractality; from a probabilistic perspective, the spectra are characterized by Pareto statistics and by Lindy Laws. The intersection of Zipf’s Law and of the GPL is also investigated, and it is shown to be: (i) the phase transition between the two markedly different fractal regimes of the GPL spectra; (ii) the phase transition between two markedly different macroscopic regimes of Zipf’s Law; (iii) Price’s Law. Metaphorically, this paper establishes navigation directions in the space of rank distributions: how to get from the ‘Zipf street’ to the ‘Price junction’, from there to the new ‘GPL avenue’, and back.

Мультиплицирование дифракционных картин от плоских геометрических фрактальных объектов с использованием метода прореживания

An algorithm is described and the features of multiplying diffraction patterns (that is, spatial frequency spectra, both calculated and experimentally obtained) from flat fractal objects using the thinning method are studied. One fractal of the L-system, described in detail and studied by the authors, was used as the main object, which greatly facilitated the interpretation of the results obtained.

Мультиплицирование дифракционных картин от плоских геометрических фрактальных объектов с использованием метода прореживания

Описан алгоритм и изучены особенности мультиплицирования дифракционных картин (то есть спектров пространственных частот, как вычисленных, так и экспериментально полученных) от плоских фрактальных объектов методом прореживания. В качестве основного объекта использовался один подробно описанный и изученный авторами фрактал L-системы, что значительно облегчало интерпретацию полученных результатов.

Shear thickening of dilute suspensions of fractal silica aggregates

The increase in viscosity under shear known as shear thickening (ST) is an inherent property of a wide variety of dense particulate suspensions. Recent studies indicate that ST systems formed by fractal particles are promising candidates for various practical applications. However, ST in fractal systems remains poorly explored. Here we experimentally study the ST behavior in suspensions of hydrophilic fumed silica (FS) particles in glycerol. Remarkably, unlike non-fractal systems, we observe a strong dependence of the onset stress for ST on the volume fraction of fractal objects and a reversible weakening of the ST response that depends strongly on the particle volume fraction as well as the properties of the FS system. Using in-situ boundary imaging, we map out the spatio-temporal flow properties during ST for different FS systems. We find that the fractal nature and structural properties like the internal branching of the particles can qualitatively explain the complex ST phase diagram of these systems.

TIME SERIES ANALYSIS USING FRACTAL THEORY

Davit Aludauri E-mail: davit.aludauri@gmail.com PhD student , Ivane Javakhishvili Tbilisi State University Tbilisi, Georgia https://orcid.org/0009-0000-7960-9668 Givi Bedianashvili E-mail: givi.bedianashvili@tsu.ge Doctor of Economic Sciences, associate professor Ivane Javakhishvili Tbilisi State University Tbilisi, Georgia https://orcid.org/0000-0003-4521-737X Giorgi Rakviashvili E-mail: giorgi.rakviashvili@iliauni.edu.ge Doctor of Mathematics, Associate Professor Ilia State University Tbilisi, Georgia https://orcid.org/0009-0003-3888-5019 Abstract. Financial markets have become a center of attention and a field of study for over a century, especially nowadays when there is an abundance of data about them, however, behavior of financial markets still stays as mystery and a field that is rife with unpredictability and uncertainty. There are many methods and techniques for analyzing time series data. Due to the fact that prices are formed by decisions made by millions of people across the planet the price in itself holds the reflection of all of those decisions which itself is a natural process. Therefore, we can look at the financial data as a fractal object and analyze them. Method that has been primarily used to analyze time series is fractal dimension which we can use to measure of the time series, in other words, we can measure how correlated is the time series and thus have an opinion about its future behavior. As we later find out, fractal dimension provides interesting insights into the time series and most importantly, it is a method that can analyze pictures and thus it is not restricted by data formats. Keywords: Uncertainty, Time series, Fractals, Fractal Dimension, Autocorrelation, Hurst Exponent JEL classification: C1, C22, C32, D81

Anatomical assessment of the trabecular structure of the alveolar bone in periodontal disease by fractal analysis method.

Early diagnosis and treatment of periodontitis, which can cause loss of bone support of the teeth, is of great importance. The use of fractal analysis method is being investigated in order to differentiate periodontal disease radiographically. Fractal analysis presents the degree of complexity in the structure of fractal objects as a numerical data, and has been used to measure changes in trabecular bone. The aim of this study was to compare the trabecular bone fractal dimension (FD) values of patients with periodontitis and gingivitis using panoramic radiographs, and to evaluate the possible relationship between age and gender with fractal dimension. Panoramic radiographs of 64 patients with gingivitis and 64 patients with periodontitis were evaluated retrospectively in the study. Using the radiographs of the patients, FD values measured from the trabecular bone were calculated with the box-counting method in the Image J programme. The FD values of both groups were compared. In addition, the relationship between age and gender parameters and FD values was evaluated within the groups. According to the results of the study, the calculated average FD value of the patients in the gingivitis group was 1.195, while the calculated average FD value of the patients in the periodontitis group was 1.196. No statistically significant difference was observed between the FD values of the gingivitis group and the periodontitis group (p > 0.05). No statistically significant correlation was observed between FD values and age and gender (p > 0.05). No statistically significant results were obtained for the calculated mean FD values of the patients in the gingivitis and periodontitis groups.

Роботизация в авиастроении

Some cses of robotized systems and complexes (RS and RC) in the aircraft industry are considered. A robotic cell for resistance spot welding is viewed, the main components of which are an industrial robot (IR), welding guns, a linear displacement module of IR, an inverting unit. Here the IR performs welding, it is equipped with welding guns as a hand. An inverting unit equipped with a stepper motor rotates the panel around the axis and sets the part to be welded within the whole welding process. The movement of the IR is also carried out due to the linear displacement module. The robotization of harnesses production is viewed, using automatic lofts and collaborative robots, which, moving along specialized lofts, with the help of a linear displacement module, install holders on the loft and ties wires into harnesses. After the operation is completed, made-up harness is put into a storage hopper. A robotized complex (RC) for manufacturing parts of the «rib» type is studied. It includes a machine equipped with a vacuum gripper, technological equipment, consisting of a cutting machine and a bending press. The technological process has two stages: cutting and bending. The cut blanks are blown out of the press onto the receiving tray. The IR performs feeding of blanks into the working area of the technological equipment and laying in the appropriate storage unit. 3D models of robotized sites are presented, their components are listed. In conclusion, the importance of analyzing the process operation is proved, when the feasibility of robotization is evident. An algorithm for analyzing the process operation is presented. It has been understood that the introduction of robotics in certain technological processes reduces the impact of hazardous production factors.

Universal scaling of the osmotic pressure for dense, quasi-two-dimensionally confined polymer melts reveals transitions between fractal dimensions.

A scaling law for the osmotic pressure of quasi-two-dimensional polymer melts as a function of concentration is obtained, which shows fractal characteristics. Structural properties such as the chains' contour length and their inner-monomer pair distribution function display fractal scaling properties as well. These predictions are confirmed with mesoscale numerical simulations. The chains are swollen and highly entangled, yet Flory's exponent is always ν = 1/2. The melt can be considered a fluid of "blobs" whose size becomes renormalized in terms of the contour's length while the fractal dimension df increases monotonically between 5/4 and 2, as the monomer concentration is increased. The semidilute scaling of the pressure is recovered when df = 1. Our results agree with recent experiments and with numerical reports on quasi-2d melts. This work provides a new paradigm to study and interpret thermodynamic and structural data in low-dimensional polymer melts, namely as fractal macromolecular objects.

An Effective Method for Compute the Roughness of Fractal Facades Based on Box-Counting Dimension (Db)

Benoit Mandelbrot coined the word “fractal” in the late 1970s, but an object is now defined as fractals in form known to artists and mathematicians for centuries. A fractal object is self-similar in that the subsections of the object are somewhat similar to the whole object. No matter how small the subdivision is, the subsection contains no less detail than the whole. Atypical example of a fractal body is the “snowflake curve” (invented by Helga von Koch (1870-1924) in 1904. There are as many relationships between architecture, the arts, and mathematics as symmetry. The golden ratio, the Fibonacci sequence in this paper explain the method of counting box and measuring the roughness ratio. And small scale analysis after calculating the box to understand fractal concepts, we must know two dimensions. Through analyzing the samples in the research, it has been proven that fractal geometry is present everywhere in our lives in nature, in buildings, and even in plants and its role in architecture is to find fractal systems that appeal to our inclinations for dynamic vitality. Therefore, finding such fractals enables us to create high-performance structures that achieve psychological, aesthetic and environmental aspects in an integrated design. Therefore, Self- Similarity Dimension (Ds) Box-counting Dimension (Db.) All of these dimensions are directly related to the fractional dimension of Mandelbrot (D). In all similar constructions there is a relationship between the scale factor and the number of the smaller pieces the original construction is divided into.

Fractal Analysis in Clinical Neurosciences: An Overview.

Over the last years, fractals have entered into the realms of clinical neurosciences. The whole brain and its components (i.e., neurons and astrocytes) have been studied as fractal objects, and even more relevant, the fractal-based quantification of the geometrical complexity of histopathological and neuroradiological images as well as neurophysiopathological time series has suggested the existence of a gradient in the pattern representation of neurological diseases. Computational fractal-based parameters have been suggested as potential diagnostic and prognostic biomarkers in different brain diseases, including brain tumors, neurodegeneration, epilepsy, demyelinating diseases, cerebrovascular malformations, and psychiatric disorders as well. This chapter and the entire third section of this book are focused on practical applications of computational fractal-based analysis into the clinical neurosciences, namely, neurology and neuropsychiatry, neuroradiology and neurosurgery, neuropathology, neuro-oncology and neurorehabilitation, neuro-ophthalmology, and cognitive neurosciences, with special emphasis on the translation of the fractal dimension and other fractal parameters as clinical biomarkers useful from bench to bedside.

Analisis Pola Harga Saham dengan Modifikasi Metode Eksponen Hurst dan Box Counting

A stock chart is a graphical representation of a stock's past performance. The dynamic pattern of stock prices is important to know because an investor wants to invest, expecting high returns with low risk. The dynamic pattern of stock prices can be known by fractal dimension analysis because the stock price graph is self-affine, which is one of the properties of fractal objects. In this study, a modification of the Hurst exponent method and box counting are used to analyze the fractal dimension. The calculated results are classified into three types, namely random, persistent and anti-persistent. Two data intervals are observed, namely January 2018-December 2021 stock prices (48 data) and January 2018-June 2022 stock prices (54 data). The resulting Hurst exponents of the two intervals are 0.043 and 0.003. Based on the resulting Hurst exponent value, the data is anti-persistent because the value 0<H<0.5. Then, the fractal dimension value obtained by applying the box-counting method is 1.547 and 1.562, meaning that Bank Rakyat Indonesia's price pattern is anti-persistent. The meaning of anti-persistent is that in certain months the stock has a high price and in the following months the stock has a low price to be traded.

EPISTEMOLOGICAL ROLE OF METHODOLOGICAL CONSTRUCTS IN FUNDAMENTAL PHYSICS

The article deals with the cognitive functions of methodological constructs in fundamental physics. The author proceeds from the fact that such universal means of cognition as a scientific picture of the world, a scientific paradigm, a research program, etc. should be considered as methodological constructs. The nature of these constructs is such that in the bosom of the methodology of science they are formed only in the mature state of scientific knowledge, turn of the 19-20th centuries. The generation of diverse scientific theories and other forms of scientific knowledge (principles, axioms, laws, concepts, etc.) led to the need for generalized means of knowledge, starting with scientific pictures of the world. Methodological constructs are difficult to define strictly, but they help to realize the human desire to see unity in diversity, build some generalized view of the diversity of fractal natural objects, and use the accumulated body of knowledge as a methodology for further scientific research. The epistemological role of methodological constructs is extremely significant for revealing not only the dynamics of the historical development of physics, but also its current state. The author concludes about the fundamental value of the diversity of methodological constructs in understanding their role in the philosophy of physics and in the development of the most fundamental theoretical physics.

Detection of lung pathology using the fractal method

<span lang="EN-US">Currently, the detection of pathology of lung cavities and their digitalization is one of the urgent problems of the healthcare industry in Kazakhstan. In this paper, the method of fractal analysis was considered to solve the task set. Diagnosis of lung pathology based on fractal analysis is an actively developing area of medical research. Conducted experiments on a set of clinical data confirm the effectiveness of the proposed methodology. The results obtained show that fractal analysis can be a useful tool for early detection of lung pathologies. It allows you to detect even minor changes in the structure and texture of lung tissues, which may not be obvious during visual analysis. The article deals with images of pathology of the pulmonary cavity, taken from an open data source. Based on the analysis of fractal objects, they were pre-assembled. Software algorithms for the operation of the information system for screening diagnostics have been developed. Based on the information contained in the fractal image of the lungs, mathematical models have been developed to create a diagnostic rule. A reference set of information features has been created that allows you to create algorithms for diagnosing the lungs: healthy and with pathologies of tuberculosis. </span>

Towering Fractals

An exponential tower is obtained by repeatedly raising a number to itself, as in xˣ, x^{x^x}, and so on. When formulated as an iterative process, it may converge to a single value, diverge to infinity, cycle between multiple values, or wander forever. Representing this property graphically leads to a beautiful fractal object. In order to plot the fractal, a method was devised to handle problems of numerical overflow and underflow that appear during the computation.

Time-of-flight spin-echo small-angle neutron scattering applied to biological cell nuclei

The organization of chromatin in the nuclei of rat lymphocyte was studied by time-of-flight spin-echo small-angle neutron scattering (ToF-SESANS). The procedures of the measurements in ToF and monochromatic SESANS modes were compared. It is shown that the sensitivity of the ToF method is significantly higher than that of the monochromatic mode. As a result, the SESANS correlation function can always be extracted from the ToF measurement of polarization, while this is not always the case with the monochromatic mode. The applicability of SESANS to fractal objects at the micrometre scale is demonstrated. The SESANS correlation function is well fitted by the exponential decay G(z) = exp(−z/ξ) with the correlation length ξ = 3.7 ± 0.1 µm. The exponential decay of the SESANS correlation function can be connected to the logarithmic correlation function γ(r) = ln(r/ξ) and the cubic law of the scattering intensity I(Q) = Q −3, which is in agreement with the concept of scattering from fractal objects. This finding is interpreted as evidence of a very specific logarithmic fractal structure of the large-scale organization of chromatin. The model of the logarithmic fractal is visualized as a hierarchical object obeying the volume-conservation principle at different scales.

RANCANGAN MOTIF KAIN POLA GARIS SIMETRIS MODEL FRAKTAL DENGAN EMPAT DAN LIMA KOMPONEN COLLAGE GARIS DAN PERSEGI-EMPAT

The motif pattern of cloths can be designed in fractal form based on Collage components composition by means of the iterated function system (IFS) encoding algorithm to encode the design position into a code and decoding algorithm to decode the code and to display the fractal object. In this paper, two kinds of Collage component composition in straight lines and self-similarity feature are discussed. The first model consists of four lines and one square shape of Collage component. The second model consists of three lines and one square shape of Collage component. The composition is made in a such way, that there are several closed planes to be colored later. The IFS code generated by the IFS encoding algorithm is normalized to ensure that the straight lines are crossed to one or two neighbor lines in the right position and the self-similarity feature is also fitted to fill the void and integrated with the lines

RANCANGAN DEKORATIF SIMETRIS BERDASARKAN MODEL FRAKTAL POLA SIERPINSKY GASKET ASIMETRIS

A symmetrical decorative design is always attractive to be used in many decoration needs. Fractal model gives a simple way in designing a decorative design that can be recycled later to have other design products easily and efficiently. The iterated function system fractal model used the code that can be modified by affine transformation operation. Sierpinsky gasket is one of a simple but attractive fractal object.