Theoretical and experimental algorithms concerning the. Fractal islamic geometric patterns based on arrangements of. This study is a pilot attempt to introduce fractal geometry to a survey mathematics 12 class. The temporal evolution of a continuous dynamical system is described by a smooth curve in the phase space, say rn. Design and analysis of fractal antennas for wideband applications. Fractal geometry defines rough or fragmented geometric shapes that can be subdivided in parts, each of which is at least approximately a reducedsize copy of the whole. Pdf this paper presents a brief overview of fractals and some possible applications of fractal geometry in architecture and civil engineering. In art circles millions of dollars are at stake concerning thirtytwo newly found works of.
Because of the limits of computer processors, all the fulldome fractal zooms stop at a magnification of 1016. Geometry, as many may know, is the study of the properties of space, and shapes. However, this thesis covers full details concerning questions about measure theory that falconer glosses over. This thesis aims to use fractal geometry for these matters, in particular the nature and building blocks of the universe.
Consider for instance, that to sample a one dimensional interval of length. The main resource in writing this thesis has been the book fractal geometry by kenneth falconer. Fractal geometry of nature in which he connected the mathematical monsters of old together into the category of fractal geometry camp, 2000. The main tool of fractal geometry is dimension in its many forms. Our understanding of fractals allows us to look back and see where assumptions were made in mathematics and science in the p. I determined the fractal dimensions for the enumerated case studies by boxcounting method. This thesis covers the foundations of fractal geometry, and gives most of the fun damental definitions and theorems that are needed to. Fractal geometry has been used in information theory, economics. This dissertation is brought to you for free and open access by the aquila digital. Strict self similarity refers to a characteristic of a form exhibited when a substructure. A ne transformation, m obius transformation, metric space, metric space of fractals, ifs, attractor, collage theorem, fractal dimension and fractal tops. This is important because fractals have proved to be vital in the description of various major components of the universe. Fractal geometry provides a promising method for representing the microscopic properties of soils and rocks. I think there are historical, sociological, and philosophical questions related to fractals that are worth investigating.
Elements of fractal geometry and dynamics yakov pesin vaughn. A remarkable property of the koch curve, which is typical of the fractals, is the. Plato, and the fractal geometry of the universe erasmus university. It was first popularized by complex renderings of terrain on a computer graphics medium. There has been much empirical evidence that the universe. The image above exists at a depth of 10176 magnification. The proposed antennas are microstrip line fed and their structure are based on fractal geometry where the resonance frequency of antenna is lowered by applying iteration techniques. Jan 16, 2001 fractal antennas and its superset fractal electrodynamics werner and mittra, 1999 is a hotbed of research activity. Of course the fractals keep going, but it becomes much slower. Some fractals, like the mandelbrot set, become even more intricate and beautiful the deeper we explore.
Fractal islamic geometric patterns based on arrangements. The purpose of this thesis is to explore fractals both topologically and computationally. An important defining property of a fractal is selfsimilarity, which refers to an infinite nesting of structure on all scales. Fractal geometry is a recent synthesis of old mathematical constructs. In this paper, a relationship between the classical theory of computation and fractal geometry is established. The illustrated wall is divided into a number of panels, each with its. Semantic scholar extracted view of plato, and the fractal geometry of the universe. We are familiar enough with the idea that a figure 0. The thesis is organized starting from some formal fractal theory in chapter 2. I searched and found a mathematical relationship between the fault statistics and fractal geometry approach. The fractal objects that we create with this process often resemble natural phenomenon. Fractal islamic geometric patterns based on arrangements of fn2gstars phil webster 49 jason st.
Therefore, the terms fractal and scaling are used interchangeably in this thesis. These structures are deterministically arising from simple dynamics of iteration of analytic functions. There are also experimental techniques capable of calculating the dimension of a given object, and these methods have proven useful in several applied areas. Fractal geometry is the geometry of the irregular shapes we find in nature which will be discussed in greater detail in section 2. Scale, fractal geometry, big data, conundrum of length, topological analysis. Pdf mandelbrot 1975 coined the term, fractal to define natural forms and the. Elements of fractal geometry and dynamics yakov pesin. A thesis submitted to the graduate faculty of auburn university. In addition, the thesis examines students making connections between mathematical topics, between mathematics and other disciplines and between mathematics and the real world. A strange attractor is a fractal, and its fractal dimension is less than the dimensions of its phase space. The proposal for this thesis, including the basic assumptions, thesis scope, evaluation criteria and expected gain is also presented. A brief history of fractal geometry properties of a fractal selfsimilarity selfsirnilarity dimension perimeter of and area under the koch curve chapter 3.
The encoding and decoding algorithms are explained through examples. Students experiences, perceptions and conceptual understandings of mathematics were studied. On the basis of the dynamics, we develop a scheme to map fractals through iterations. The definition of a fractal porous medium, as given in rieu and sposito 1991, is a medium whose pore space and solid matrix can be characterized by statistical selfsimilarity. Fractals in architecture abstract the fractal geometry has emerged from the chaos theory within the new scientific group including complex theory, which has given a new vision to the natural global system, that is considered far from regularity, order, linearity, and the power of prediction which considered the euclidean geometry.
We hope to demystify the concepts of fractal geometry and make it available to all who are interested. Pdf fractal geometry and applicationsan introduction to. This chapter concludes with a brief description of the content of each of the following chapters. Methods of classical geometry and calculus are unsuited to studying fractals and we need alternative techniques. Fractal geometry deal with the concept of selfsimilarity and roughness in the nature. Deterministic fractals like julia sets are crucial for understanding the fractal phenomenon. Feb 07, 2018 undecidable problems in fractal geometry simant dube department ofmathematics, statistics, and computing science, university of new england, armidale nsw 2351, australia abstract. Two numerical versions of asymmetric structure were created based on fractal geometry, as well as a symmetric structure that was taken as a reference. Assessing awareness, interest, and knowledge of fractal geometry. Some of the wellknown fractal properties of brownian motion are mentioned and some are proved using nonstandard noii. Thereby, the general approach and much content, like example 2. Pdf fractal geometry and applicationsan introduction.
Fractal trading pdf by soul esprit they are more bars required for, posting your methodology you put up the guesswork out. What are the philosophical implications of fractal geometry. The focus of this thesis is on how fractal geometry can be used in applications to computer graphics or to model natural objects. Various structures of fractals are going to be tested in order to achieve a comparison between them.
It also overcomes the drawbacks of traditional euclidean thinking and provides deeper insights for giscience research in the big data era. Students making connections through interactions with fractal. A ne transformation, m obius transformation, metric space, metric space of fractals, ifs, attractor, collage theorem, fractal dimension and fractal. This thesis describes the influence of an asymmetric structure on the vibration response of satellites, specifically of cubesat. The purpose of this work is to present fractal geometry to the graphics programmer as a simple workable technique. Hank i still has not as, confirmation indicator always use them like. The thesis relies on the third definition of fractal geometry. The fourth chapter applies the previous three to the study of the fractal geometry of brownian motion.
A set or pattern is fractal if the scaling of far more small things than large ones recurs multiple times. First, we have numerically estimated the subset fractal dimensions d s describing the scaling of some subsets s of the fractal cluster with the linear cluster size r. Pdf plato, and the fractal geometry of the universe semantic. This thesis comprises analytic and numerical studies of static, geometrical properties of fractals and dynamical processes in them. Fractal geometry of wood internal surfaces in the hygroscopic range by bingye hao b. Fractal antennas for wireless communication systems. This thesis covers the foundations of fractal geometry, and gives most of the fun. Lapidus published fractal geometry and applications an introduction to this volume find, read and cite all the research you need on researchgate. You may answer the assignment questions working in pairs and providing on set of. A brief fractal audio model is provided based on the conventional fractal image model. A list of necessary support equipment, both hardware and software is included.
Abstract the use of fractal geometry has helped in the description and quantification of patterns in plants, from an individual plant root system to plant species geographical levels of distribution. Studying students sense making of fractal geometry sfus summit. Fast track paper the fractal geometry of the surface. Pdf fractals and fractal design in architecture semantic. Students making connections through interactions with. For centuries, this branch of mathematics was used to describe the physical world. Lorenz, fractals and fractal architecture, from url.
In short, irregular details or patterns are repeated themselves in even smaller scale. They were constructed as simple counterexamples to commonly held notions in set theory, analysis and geometry. Dynamic characterization of naturally fractured reservoirs with fractal geometry. Introduction to fractal geometry 3 fractal composite leaf 4 dynamical systems. From this basic thesis, flake explores what he considers to be. In art circles millions of dollars are at stake concerning thirtytwo newly found works of art reportedly by jackson pollock. Characterization of the canawoodford shale using fractal based, stochastic inversion canadian county, oklahoma approved by supervising committee.
Fractal geometry, graph and tree constructions diva. Introduction to fractal geometry 7 we have therefore an example of a curve for which the lengths of the approximating curves diverge like the coast of great britain, and which is full of angular points, like the trajectory of the brownian motion. Fractal thinking effectively describes the characteristics of geographic big data across all scales. In the research journals, we see reports of active research covering such diverse areas of fractal electrodynamics as the study of scattering from fractal surfaces a signature of the surface is imprinted within the scattered. The thesis committee for barry michael borgman certifies that this is the approved version of the following thesis. Fractal geometry mathematical foundations and applications. First, we have numerically estimated the subset fractal dimensions d s describing the scaling of some subsets s of the fractal cluster with the linear cluster size r in the qstate potts models. Mandelbrot credits computers with the awakening of experimental mathematics peitgen, jurgens, and saupe, 1992, 9. This thesis investigates grade nine students engagement in interactive sessions in regard to fractal geometry. Its rapid acceptance has been achieved due to its ability to model phenomena. A step further was viewing the problem by the angle of fractal geometry.
It was first thought that euclidean geometry is the only type of geometry we can. Golnaz mohammadi university of washington thesis proposal. In the third chapter of this thesis we will focus on calculating the hausdorff dimension of some selfsimilar fractals in a rigorous mathematical way, but before that, we would like to show some examples of fractal structures that can be found in nature, architecture and some other. This is the idea that certain geometrical features in a medium. Figure 1 exemplifies the use of geometry and geometric patterns in islamic architecture.
Since the fractal dimension, d, is a very popular and arguably important parameter in describing the fractal rough surfaces, four different methods are implemented in calculating the value and these four methods are then compared. However, raw quantitative data, such as fractal dimension values can be correlated with the size or spacefilling of organisms under investigation. An analysis of the multiscale structure of rough surfaces. Fractal geometry may help us to understand and analyze complexity that can be found in towns of the middle ages but also in cathedrals and other manmade object s up to these days. Fractal geometry has since spawned research in many diverse scientific disciplines. An application i frequent the fxcm trading indicators and other indicators. Dynamic characterization of naturally fractured reservoirs. In 1975, he used the word fractal latin for broken to denote an object whose hausdorff dimension a measure of roughness or chaos is larger than the topological one. July 17, 2014 classical geometry, which traces its origins back to euclid, is concerned with figures, shapes and the properties of space. Fractal geometry is also used to document the authenticity of art.
Methodology of analysis by lauro jesus vargas munoz a thesis submitted to the faculty of graduate studies in partial fulfilment of the requirements for the degree of master of science graduate program in chemical and petroleum engineering calgary, alberta july, 2016. Case studies on fractal and topological analyses of. This thesis aims to provide some elucidation on these matters, in particular with the use of a new branch of mathematics called fractal geometry. Wide band fractal antennas geometry has been proposed in this thesis. You may answer the assignment questions working in pairs and providing on set of answers containing both of your names. For our purposes, fractals will come from certain dynamical systems, and will lead us to an understanding of certain aspects of chaos. Mathematical foundations and applications and numerous book.
1538 1231 374 1593 1526 368 112 1305 698 144 336 174 190 52 1257 624 1586 1435 1248 1314 327 951 275 1409 702 605 147 1250 1171 1210 293 1266 671 808 802 373 1520 45