Principles Of Fractal Geometry Research Articles

Fractal networks: Topology, dimension, and complexity.

Over the past two decades, the study of self-similarity and fractality in discrete structures, particularly complex networks, has gained momentum. This surge of interest is fueled by the theoretical developments within the theory of complex networks and the practical demands of real-world applications. Nonetheless, translating the principles of fractal geometry from the domain of general topology, dealing with continuous or infinite objects, to finite structures in a mathematically rigorous way poses a formidable challenge. In this paper, we overview such a theory that allows to identify and analyze fractal networks through the innate methodologies of graph theory and combinatorics. It establishes the direct graph-theoretical analogs of topological (Lebesgue) and fractal (Hausdorff) dimensions in a way that naturally links them to combinatorial parameters that have been studied within the realm of graph theory for decades. This allows to demonstrate that the self-similarity in networks is defined by the patterns of intersection among densely connected network communities. Moreover, the theory bridges discrete and continuous definitions by demonstrating how the combinatorial characterization of Lebesgue dimension via graph representation by its subsets (subgraphs/communities) extends to general topological spaces. Using this framework, we rigorously define fractal networks and connect their properties with established combinatorial concepts, such as graph colorings and descriptive complexity. The theoretical framework surveyed here sets a foundation for applications to real-life networks and future studies of fractal characteristics of complex networks using combinatorial methods and algorithms.

Fractals in the Neurosciences: A Translational Geographical Approach.

The chapter presents three new fractal indices (fractal fragmentation index, fractal tentacularity index, and fractal anisotropy index) and normalized Kolmogorov complexity with proven applicability in geographic research, developed by the authors, and the possibility of their future use in neuroscience. The research demonstrates the relevance of fractal analysis in different fields and the basic concepts and principles of fractal geometry being sufficient for the development of models relevant to the studied reality. Also, the research highlighted the need to continue interdisciplinary research based on known fractal indicators, as well as the development of new analysis methods with the translational potential between fields.

Box-Counting Fractal Analysis: A Primer for the Clinician.

This chapter lays out the elementary principles of fractal geometry underpinning much of the rest of this book. It assumes a minimal mathematical background, defines the key principles and terms in context, and outlines the basics of a fractal analysis method known as box counting and how it is used to perform fractal, lacunarity, and multifractal analyses. As a standalone reference, this chapter grounds the reader to be able to understand, evaluate, and apply essential methods to appreciate and heal the exquisitely detailed fractal geometry of the brain.

Fractal Geometry in High-Frequency Trading: Modeling Market Microstructure and Price Dynamics

This theoretical article delves into the intricate world of high-frequency trading (HFT) without empirical testing of real-world data, focusing on the incorporation of fractal geometry principles to enhance our understanding of market microstructure and price dynamics. In the introduction, we outline the significance of this research in the context of modern financial markets and lay out the objectives of our theoretical analysis. The article then takes an in-depth dive into fractal geometry fundamentals, illuminating its core concepts and its relevance within financial markets. Subsequently, the article explores the landscape of high-frequency trading, offering an overview of this dynamic domain and how fractal geometry can be incorporated into trading models. The section on modeling market microstructure presents theoretical approaches to understanding order flow dynamics, including novel derivations and equations. It then transitions into fractal-based approaches for analyzing the complexities of market microstructure, providing both an original perspective and numbered equations. Moreover, this article investigates the theoretical modeling of price dynamics, underscoring the pivotal role of fractal geometry in enriching these models. The discussion revolves around the fundamental autoregressive models and multifractal models, and it elucidates how fractal geometry principles, such as the Hurst exponent, come into play. We explore the self- similarity of price dynamics, fractal dimensions, and how these aspects can be integrated into high-frequency trading strategies. Overall, this article offers a comprehensive theoretical exploration of fractal geometry's implications in the realm of high-frequency trading, providing valuable insights for both researchers and practitioners seeking to fathom the complexities of market microstructure and price dynamics. The incorporation of fractal principles into financial models fosters a deeper understanding of self-similarity and complexity within financial markets, even in the absence of empirical data.

프랙탈 기하학 차원에서 본 자하 하디드 건축작품의 공간형태특성에 관한 연구

This study utilizes fractal geometry, which is evaluated as a new alternative in the field of science, to interpret and apply complex and diverse architectural phenomena in the field of architectural design to create multi-dimensional space forms and respond to users' unpredictable and flexible demands. We want to explore the possibilities of whether it can be used with various algorithms that can be used.
 In this process, we tried to understand the concept, characteristics, and compositional principles of fractal geometry, and to interpret the creation process of spatial forms in Zaha Hadid's architectural works erected after the 2000s from the perspective of fractal geometry. In addition, we tried to explore the possibility of application to contemporary architecture by proposing the aspects of the architect's fractal characteristics through diagrams and analyzing the proportion of utilization of various algorithms to propose them as a mechanism for creating spatial forms.

Effect of wet ball milling on copper ore flotation by fractal geometry

ABSTRACT One of the important parameters in mineral processing is particles morphology, which is a function of the comminution mechanism. Particles morphology has considerable impacts in flotation process. Variations in particles morphology have been studied by different methods, including fractal geometry. This study aims to investigate the ball milling effect on particles flotation using fractal dimension, in which the calculations in the particles morphology has been made using the step-length method. The ore sample of under-investigation was a copper ore. The examined parameters include grinding time, ball load, and solid weight percentage. The collected fractal dimensions for particles ranged from 1 to 2, which is according to the principle of fractal geometry. Based on the results, low (30 minute) or high (50 minute) grinding time reduces fractal dimension and recovery because low grinding time is resulted in a coarse grain product while high grinding time is resulted in fine particles with spherical shapes. If balls load is high (8710 gr), predominant comminution force turns to an abrasive force are resulted in fine and spherical particles due to an increase in the accumulation volume. When the ball load decreased (4690 gr), the coarse particles were produced. Fractal dimensions and recovery are decreased in both cases. Low (40%) and high (60%) solid weight percentages are resulted in ineffective comminution, producing a coarse grain product. As a result, fractal dimensions and recovery are decreased. There is a direct correlation between recovery and fractal dimension.

Nietzsche and Fractal Geometry: a philosophical continuity

The purpose of this work is to highlight the epistemological proximity between Nietzsche’s philosophy of science and the underlying philosophical principles of fractal geometry, as illustrated in the main work of its creator, French mathematician Benoit Mandelbrot.This work also aims to find the end of this philosophical continuity, finding an important divergence between Nietzsche’s philosophy of risk taking and Mandelbrot’s legacy in risk management.

Maple Leaf Inspired Conductive Fiber with Hierarchical Wrinkles for Highly Stretchable and Integratable Electronics.

Stretchable and durable conductors are significant to the development of wearable devices, robots, human-machine interfaces, and other artificial intelligence products. However, the desirable strain-insensitive conductivity and low hysteresis are restricted by the failure of stretchable structures and mismatch of mechanical properties (rigid conductive layer and elastic core substrate) under large deformation. Here, based on the principles of fractal geometry, a stretchable conductive fiber with hierarchical wrinkles inspired by the unique shape of the maple leaf was fabricated by combining surface modification, interfacial polymerization, and improved prestrain finishing methods to break through this dilemma. The shape and size of wrinkles predicted by buckling analysis via the finite element method fit well with that of actual wrinkles (30-80 μm of macro wrinkles and 4-6 μm of micro wrinkles) on the fabricated fiber. Such hierarchically wrinkled conductive fiber (HWCF) exhibited not only excellent strain-insensitive conductivity denoted by the relative resistance change ΔR/R0 = 0.66 with R0 the original resistance and ΔR the change of resistance after the concrete strain reaching up to 600%, but also low hysteresis (0.04) calculated by the difference in area between stretching and releasing curve of the ΔR/R0 strain under 300% strain and long-term durability (>1000 stretching-releasing cycles). Furthermore, the elastic conductive fiber with such a bionic structure design can be applied as highly stretchable electrical circuits for illumination and monitors for the human motion under large strains through tiny and rapid resistance changes as well. Such a smart biomimetic material holds great prospects in the field of stretchable electronics.

Development of Body Painting 3D Printing Techniques Using Fractal Design

With development of advanced technologies, the field of beauty is under strong pressure to try new approaches in line with the highly increasing interest in advance 3D printing based on 3D graphic design data. An increasing number of researches have been conducted to develop beauty and art design object using computer design programs. As part of this, three works were presented in this paper in which body painting designs and object were applied to mannequins by producing them with help of 3D printing techniques based on the motive of fractal that started from nature. This study examined how the generation principle of fractal geometry appears in the form of nature. The generation principle of fractal geometry models nature, fibonacci, and crystalline pattern by non-linearity, irregularity, and randomness around the iterative rule of self similarity. The present study is thought to be meaningful in that it suggests the possibility and practical value of a design method that can be technically and easily accessible to those majoring in beauty by means of its utility as a low-end 3D printing object.

A Review of the Fractal Market Hypothesis for Trading and Market Price Prediction

This paper provides a review of the Fractal Market Hypothesis (FMH) focusing on financial times series analysis. In order to put the FMH into a broader perspective, the Random Walk and Efficient Market Hypotheses are considered together with the basic principles of fractal geometry. After exploring the historical developments associated with different financial hypotheses, an overview of the basic mathematical modelling is provided. The principal goal of this paper is to consider the intrinsic scaling properties that are characteristic for each hypothesis. In regard to the FMH, it is explained why a financial time series can be taken to be characterised by a 1/t1−1/γ scaling law, where γ>0 is the Lévy index, which is able to quantify the likelihood of extreme changes in price differences occurring (or otherwise). In this context, the paper explores how the Lévy index, coupled with other metrics, such as the Lyapunov Exponent and the Volatility, can be combined to provide long-term forecasts. Using these forecasts as a quantification for risk assessment, short-term price predictions are considered using a machine learning approach to evolve a nonlinear formula that simulates price values. A short case study is presented which reports on the use of this approach to forecast Bitcoin exchange rate values.

ПРОЯВЛЕНИЕ ФРАКТАЛЬНОСТИ В АРХИТЕКТУРЕ ЗДАНИЙ И СООРУЖЕНИЙ

the article shows the presence of proportional correspondence of parts and the whole using the principles of fractal geometry in the formation of the concept of beauty of natural objects and phenomena. The objects of the artificial environment for the formation of images on the example of mathematical models of calculation of natural objects that have the properties of fractal are investigated. The article presents the types of fractals. Examples of fractal in physics, biology, chemistry are given. Fractal geometry can and should be used in modeling the structure of the artificial environment that humanity creates through architecture. The very process of designing a single building, and its integration into the urban environment, which in the process of formation is also quite chaotic, falls under the principles of fractal geometry. Modeling of these processes using the principles of fractal geometry allows not only to create the most comfortable artificial urban environment, but also to monitor the flow of all processes within it.

Cross-scale characteristics of backfill material using NMR and fractal theory

Abstract This paper analyzed the pore structure, quantified the pore fractal dimension, calculated the grading index (GI) of mixed aggregate, and studied the relationship among GI, pore structure, and strength to describe the cross-scale characteristics of backfill, which is made from stone powder and cemented tailing. A series of experiments were conducted on stone powder cement tailings backfill (SPCTB). The GI formulas for mixed aggregates, containing stone powder and tailings, were derived based on the Fuller theory. The nuclear magnetic resonance (NMR) fractal dimensions of backfills were derived using fractal geometry principles. Compared to the mesopore and macropore fractal dimensions, the correlation between micropore fractal dimension and macro-properties in terms of NMR porosity, pore structure complexity, uniaxial compression strength (UCS), and GI is the most significant. Macropore fractal dimension is generally correlated with UCS and GI and the other properties such as the shape of mixed aggregates also have an impact on fractal dimension. However, mesopore fractal dimension has no obvious relationship with macro-properties. Finally, the relationship between GI and UCS was studied, which contributed to improving backfill’s strength and optimizing gradation.

Evolution of the fractal structure

The principles of fractal geometry are applicable in the development of new designs based on fractal shaping and the evolution of fractal structures. It’s necessary to use new materials to design fractal structure. The evolution of the structure is characterized by optimization of the initial geometry in the process of iteration. Thus, the fractal structures of older generations from innovative material have the best indicators of reliability. The aim of this study is to develop new designs using the principles of fractal shaping. To reduce the weight of the elements of the building frame in parallel with the evolution of the structure, it is necessary to explore innovative materials with certain material properties. The development of fractal structures requires the creation of new technologies and materials.

The manifestation of fractality in the architecture of buildings and structures

The article shows the presence of proportional correspondence of the parts and the whole using the principles of fractal geometry in the natural objects and phenomena beauty concept formation. The objects of the artificial environment are studied for the formation of images using the mathematical models’ example for calculating natural objects with fractal properties.

Surface Morphology: Theoretical Model and Correlation with Experimental Results

In this work, we propose a model for the description of the surface morphogenesis of a dispersed system of the solid-solid type. To obtain the model, stochastic formalism based on the master equation and the principles of fractal geometry was applied, so that the surface morphology is characterized by the fractal dimension and the roughness exponent, which are expressed as a function of the composition of the dispersed system and the dynamic parameters associated with surface formation. Theoretical results obtained were compared with experimental results, finding that the variable that shows a significant effect on the morphology of the surface of the solid-solid dispersed system is the specific surface area of the particles of the dispersed phase found in the surface, as predict theoretically.

프랙탈 기하학의 조형원리를 적용한 3D 프린팅 패션소재 연구 - 패션소재의 기본조직 응용을 중심으로 -

프랙탈 기하학의 조형원리를 적용한 3D 프린팅 패션소재 연구 - 패션소재의 기본조직 응용을 중심으로 -

Stochastic modeling of asphaltenes deposition and prediction of its influence on friction pressure drop

When dealing with heavy and extra-heavy crude oils, petroleum industry faces the deposition and incrustation of solids on pipelines walls during fluids transportation. Such a deposition phenomenon is supposed to be caused by asphaltenes aggregation. In this work, is presented a stochastic model that predicts the behavior of friction pressure drop with respect to asphaltenes concentration and the deposition and detaching velocities through the principles of fractal geometry and differential fractional calculus. Results show that at higher values of asphaltenes total concentration and deposition velocity, the friction pressure losses are also increased, which is an expected behavior because of the thickness and morphology of the asphaltenes deposited layer.

Evaluation of the Percolation Sensitivity of Loose Sandstone Using Digital Core Technology

Background:The integrity of the extracted core in loose sandstone gas reservoirs is poor, and because hydration and collapse easily occur, it is difficult to evaluate the sensitivity characteristics accurately by the traditional core flooding experiments.Objectives:We instead investigate the stress sensitivity and water sensitivity of the formation water soaking time using digital core technology.Methods:We take the core of a loose sandstone gas reservoir as a research object and begin by scanning the core samples with a CT scanner. A three-dimensional image of the core can be obtained, the digital information extracted, the pore structure of the porous media mapped directly to the network, and a digital core established using the principles of fractal geometry. The three-dimensional pore network model can also be extracted. Next, we can compare and correct the results calculated by the model based on the real core experimental results, and an objective and effective digital core model can be obtained.Results and Conclusion:Finally, we can calculate the different effective stress, pore throat parameters (pore throat radius, shape factor, coordination number, pore-throat ratio) and relative permeability of different formation water injury times. The research results demonstrate that in sandstone gas reservoir development, as the effective stress continuously increases, the rock pore-throat parameters continue to decrease, and the permeability of the reservoir rock ultimately declines by more than 43.2%. Clay minerals will expand after the edge and bottom water intrude into the reservoir and soak it for a long time: the pore throat is significantly narrowed within 30 days, while after 30 days more, the pore throat undergoes any only slight further changes, and the final permeability decline of the reservoir rock is up to 5.7%. The research results provide important basic petrophysical data for the development of loose sandstone gas reservoirs which, in turn, provide a scientific basis for formulating a reasonable gas production rate in a gas reservoir.

Textile Design by Principle of Fractal Geometry

Textile Design by Principle of Fractal Geometry

Quantitative angiography methods for bifurcation lesions: a consensus statement update from the European Bifurcation Club.

Bifurcation lesions represent one of the most challenging lesion subsets in interventional cardiology. The European Bifurcation Club (EBC) is an academic consortium whose goal has been to assess and recommend the appropriate strategies to manage bifurcation lesions. The quantitative coronary angiography (QCA) methods for the evaluation of bifurcation lesions have been subject to extensive research. Single-vessel QCA has been shown to be inaccurate for the assessment of bifurcation lesion dimensions. For this reason, dedicated bifurcation software has been developed and validated. These software packages apply the principles of fractal geometry to address the "step-down" in the bifurcation and to estimate vessel diameter accurately. This consensus update provides recommendations on the QCA analysis and reporting of bifurcation lesions based on the most recent scientific evidence from in vitro and in vivo studies and delineates future advances in the field of QCA dedicated bifurcation analysis.