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Fractals and the Chaos Game
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2006 Fractals and the Chaos Game Stacie Lefler University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Lefler, Stacie, "Fractals and the Chaos Game" (2006). MAT Exam Expository Papers. 24. https://digitalcommons.unl.edu/mathmidexppap/24 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Fractals and the Chaos Game Expository Paper Stacie Lefler In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. David Fowler, Advisor July 2006 Lefler – MAT Expository Paper - 1 1 Fractals and the Chaos Game A. Fractal History The idea of fractals is relatively new, but their roots date back to 19 th century mathematics. A fractal is a mathematically generated pattern that is reproducible at any magnification or reduction and the reproduction looks just like the original, or at least has a similar structure. Georg Cantor (1845-1918) founded set theory and introduced the concept of infinite numbers with his discovery of cardinal numbers. He gave examples of subsets of the real line with unusual properties. These Cantor sets are now recognized as fractals, with the most famous being the Cantor Square . -
Strictly Self-Similar Fractals Composed of Star-Polygons That Are Attractors of Iterated Function Systems
Strictly self-similar fractals composed of star-polygons that are attractors of Iterated Function Systems. Vassil Tzanov February 6, 2015 Abstract In this paper are investigated strictly self-similar fractals that are composed of an infinite number of regular star-polygons, also known as Sierpinski n-gons, n-flakes or polyflakes. Construction scheme for Sierpinsky n-gon and n-flake is presented where the dimensions of the Sierpinsky -gon and the -flake are computed to be 1 and 2, respectively. These fractals are put1 in a general context1 and Iterated Function Systems are applied for the visualisation of the geometric iterations of the initial polygons, as well as the visualisation of sets of points that lie on the attractors of the IFS generated by random walks. Moreover, it is shown that well known fractals represent isolated cases of the presented generalisation. The IFS programming code is given, hence it can be used for further investigations. arXiv:1502.01384v1 [math.DS] 4 Feb 2015 1 1 Introduction - the Cantor set and regular star-polygonal attractors A classic example of a strictly self-similar fractal that can be constructed by Iterated Function System is the Cantor Set [1]. Let us have the interval E = [ 1; 1] and the contracting maps − k k 1 S1;S2 : R R, S1(x) = x=3 2=3, S2 = x=3 + 2=3, where x E. Also S : S(S − (E)) = k 0 ! − 2 S (E), S (E) = E, where S(E) = S1(E) S2(E). Thus if we iterate the map S infinitely many times this will result in the well known Cantor[ Set; see figure 1. -
IJR-1, Mathematics for All ... Syed Samsul Alam
January 31, 2015 [IISRR-International Journal of Research ] MATHEMATICS FOR ALL AND FOREVER Prof. Syed Samsul Alam Former Vice-Chancellor Alaih University, Kolkata, India; Former Professor & Head, Department of Mathematics, IIT Kharagpur; Ch. Md Koya chair Professor, Mahatma Gandhi University, Kottayam, Kerala , Dr. S. N. Alam Assistant Professor, Department of Metallurgical and Materials Engineering, National Institute of Technology Rourkela, Rourkela, India This article briefly summarizes the journey of mathematics. The subject is expanding at a fast rate Abstract and it sometimes makes it essential to look back into the history of this marvelous subject. The pillars of this subject and their contributions have been briefly studied here. Since early civilization, mathematics has helped mankind solve very complicated problems. Mathematics has been a common language which has united mankind. Mathematics has been the heart of our education system right from the school level. Creating interest in this subject and making it friendlier to students’ right from early ages is essential. Understanding the subject as well as its history are both equally important. This article briefly discusses the ancient, the medieval, and the present age of mathematics and some notable mathematicians who belonged to these periods. Mathematics is the abstract study of different areas that include, but not limited to, numbers, 1.Introduction quantity, space, structure, and change. In other words, it is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Mathematicians seek out patterns and formulate new conjectures. They resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity. -
New Thinking About Math Infinity by Alister “Mike Smith” Wilson
New thinking about math infinity by Alister “Mike Smith” Wilson (My understanding about some historical ideas in math infinity and my contributions to the subject) For basic hyperoperation awareness, try to work out 3^^3, 3^^^3, 3^^^^3 to get some intuition about the patterns I’ll be discussing below. Also, if you understand Graham’s number construction that can help as well. However, this paper is mostly philosophical. So far as I am aware I am the first to define Nopt structures. Maybe there are several reasons for this: (1) Recursive structures can be defined by computer programs, functional powers and related fast-growing hierarchies, recurrence relations and transfinite ordinal numbers. (2) There has up to now, been no call for a geometric representation of numbers related to the Ackermann numbers. The idea of Minimal Symbolic Notation and using MSN as a sequential abstract data type, each term derived from previous terms is a new idea. Summarising my work, I can outline some of the new ideas: (1) Mixed hyperoperation numbers form interesting pattern numbers. (2) I described a new method (butdj) for coloring Catalan number trees the butdj coloring method has standard tree-representation and an original block-diagram visualisation method. (3) I gave two, original, complicated formulae for the first couple of non-trivial terms of the well-known standard FGH (fast-growing hierarchy). (4) I gave a new method (CSD) for representing these kinds of complicated formulae and clarified some technical difficulties with the standard FGH with the help of CSD notation. (5) I discovered and described a “substitution paradox” that occurs in natural examples from the FGH, and an appropriate resolution to the paradox. -
Herramientas Para Construir Mundos Vida Artificial I
HERRAMIENTAS PARA CONSTRUIR MUNDOS VIDA ARTIFICIAL I Á E G B s un libro de texto sobre temas que explico habitualmente en las asignaturas Vida Artificial y Computación Evolutiva, de la carrera Ingeniería de Iistemas; compilado de una manera personal, pues lo Eoriento a explicar herramientas conocidas de matemáticas y computación que sirven para crear complejidad, y añado experiencias propias y de mis estudiantes. Las herramientas que se explican en el libro son: Realimentación: al conectar las salidas de un sistema para que afecten a sus propias entradas se producen bucles de realimentación que cambian por completo el comportamiento del sistema. Fractales: son objetos matemáticos de muy alta complejidad aparente, pero cuyo algoritmo subyacente es muy simple. Caos: sistemas dinámicos cuyo algoritmo es determinista y perfectamen- te conocido pero que, a pesar de ello, su comportamiento futuro no se puede predecir. Leyes de potencias: sistemas que producen eventos con una distribución de probabilidad de cola gruesa, donde típicamente un 20% de los eventos contribuyen en un 80% al fenómeno bajo estudio. Estos cuatro conceptos (realimentaciones, fractales, caos y leyes de po- tencia) están fuertemente asociados entre sí, y son los generadores básicos de complejidad. Algoritmos evolutivos: si un sistema alcanza la complejidad suficiente (usando las herramientas anteriores) para ser capaz de sacar copias de sí mismo, entonces es inevitable que también aparezca la evolución. Teoría de juegos: solo se da una introducción suficiente para entender que la cooperación entre individuos puede emerger incluso cuando las inte- racciones entre ellos se dan en términos competitivos. Autómatas celulares: cuando hay una población de individuos similares que cooperan entre sí comunicándose localmente, en- tonces emergen fenómenos a nivel social, que son mucho más complejos todavía, como la capacidad de cómputo universal y la capacidad de autocopia. -
The Chaos Game
Complexity is around us. Part one: the chaos game Dawid Lubiszewski Complex phenomena – like structures or processes – are intriguing scientists around the world. There are many reasons why complexity is a popular topic of research but I am going to describe just three of them. The first one seems to be very simple and says “we live in a complex world”. However complex phenomena not only exist in our environment but also inside us. Probably due to our brain with millions of neurons and many more neuronal connections we are the most complex things in the universe. Therefore by understanding complex phenomena we can better understand ourselves. On the other hand studying complexity can be very interesting because it surprises scientists at least in two ways. The first way is connected with the moment of discovery. When scientists find something new e.g. life-like patterns in John Conway’s famous cellular automata The Game of Life (Gardner 1970) they find it surprising: In Conway's own words, When we first tracked the R-pentamino... some guy suddenly said "Come over here, there's a piece that's walking!" We came over and found the figure...(Ilachinski 2001). However the phenomenon of surprise is not only restricted to scientists who study something new. Complexity can be surprising due to its chaotic nature. It is a well known phenomenon, but restricted to a special class of complex systems and it is called the butterfly effect. It happens when changing minor details in the system has major impacts on its behavior (Smith 2007). -
Information to Users
INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9427736 Exploring the computational capabilities of recurrent neural networks Kolen, John Frederick, Ph.D. The Ohio State University, 1994 Copyright ©1994 by Kolen, John Frederick. -
Bridges Conference Proceedings Guidelines
Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture An Introduction to Leaping Iterated Function Systems Mingjang Chen Center for General Education National Chiao Tung University 1001 University Rd. Hsinchu, Taiwan E-mail: [email protected] Abstract The concept of “Leaping Iterated Function Systems (LIFS in short)” is a variation of Iterated Function Systems (IFS), that originated from “self-similarity”, i.e., the whole has the same shape as one or more of the parts. The methodology of Leaping IFS is first to construct a structure of the whole by parts, then to convert it to an image, and then to replace each part with this image. Repeat the procedures until the outcome is visually satisfied. One of the key features of Leaping IFS is that computer resource consumption will be under control during the iterations because the number of parts in the whole structure is fixed. Introduction Iterated Function Systems (IFS) is an interactive method of constructing fractals. First proposed by John Hutchinson [2] in 1981, Michael Barnsley [1] then applied this method to imitate self-similar patterns in the real-world. Translation, rotation, scaling, reflection, and shearing are included in the function system of linear fractals; their relative positions, orientations and scaling of geometry elements are used to define 2D geometrical transformation visually. In particular, the concept of the Multiple Reduction Copy Machine algorithm (MRCM) was used [6] to introduce IFS. An interactive IFS Fractal generator based on “the Collage Theorem" allows users to sketch first an approximate outline of the desired fractal, and then cover it with deformed images of itself to achieve the collage and then render the attractor. -
Julia Fractals in Postscript Fractal Geometry II
Kees van der Laan EUROTEX 2012 & 6CM PROCEEDINGS 47 Julia fractals in PostScript Fractal Geometry II In memory of Hans Lauwerier Abstract Lauwerier’s BASIC codes for visualization of the usual Julia fractals: JULIAMC, JULIABS, JULIAF, JULIAD, JULIAP, of the Mandelbrot fractal MANDELx, MANDIS, MANDET and his codes for the advanced circular symmetric Julia fractals JULIAS, JULIASYMm, JULIASYM, FRACSYMm, as well as the classical 1D bifurcation picture Collet, have been converted into PostScript defs. Examples of use are included. A glimpse into Chaos theory, in order to understand the principles and peculiarities underlying Julia sets, is given. Bifurcation diagrams of the Ver- hulst model of limited growth and of the Julia quadratic dynamical system — M-fractal — have been included. Barnsley’s triples: fractal, IFS and equivalent dynamical system are in- troduced. How to use the beginnings of colours in PostScript is explained. How to obtain Julia fractals via Stuif’s Julia fractal viewer, and via the special fractal packages Winfract, XaoS, and Fractalus is dealt with. From BASIC codes to PostScript library defs entails soft- ware engineering skills. The paper exhibits experimental fractal geometry, practical use of minimal TEX, as well as ample EPSF programming, and is the result of my next step in ac- quainting myself with Lauwerier’s 10+ years work on fractals. Keywords Acrobat Pro, Adobe, art, attractor, backtracking, Barnsley, BASIC, bifurcation, Cauchy con- vergence, chaos, circle symmetry, dynamical systems, EPSF, escape-time algorithm, Feigen- baum constant, fractal dimension D, Fractalus package, FIFO, fractal geometry, IDE (Inte- grated development Environment), IFS (Iterated Function System), Julia, Lauwerier, Man- delbrot, mathematical software, minimal encapsulated PostScript, minimal plain TeX, Monte Carlo, μ-geometry, periodic doubling, Photoshop, PSlib, PSView, repeller, self-similarity, software engineering, Stuif’s previewer, TEXworks, (adaptable) user space, Verhulst growth model, Winfract package, XaoS fractal package. -
Martin Gardner Papers SC0647
http://oac.cdlib.org/findaid/ark:/13030/kt6s20356s No online items Guide to the Martin Gardner Papers SC0647 Daniel Hartwig & Jenny Johnson Department of Special Collections and University Archives October 2008 Green Library 557 Escondido Mall Stanford 94305-6064 [email protected] URL: http://library.stanford.edu/spc Note This encoded finding aid is compliant with Stanford EAD Best Practice Guidelines, Version 1.0. Guide to the Martin Gardner SC064712473 1 Papers SC0647 Language of Material: English Contributing Institution: Department of Special Collections and University Archives Title: Martin Gardner papers Creator: Gardner, Martin Identifier/Call Number: SC0647 Identifier/Call Number: 12473 Physical Description: 63.5 Linear Feet Date (inclusive): 1957-1997 Abstract: These papers pertain to his interest in mathematics and consist of files relating to his SCIENTIFIC AMERICAN mathematical games column (1957-1986) and subject files on recreational mathematics. Papers include correspondence, notes, clippings, and articles, with some examples of puzzle toys. Correspondents include Dmitri A. Borgmann, John H. Conway, H. S. M Coxeter, Persi Diaconis, Solomon W Golomb, Richard K.Guy, David A. Klarner, Donald Ervin Knuth, Harry Lindgren, Doris Schattschneider, Jerry Slocum, Charles W.Trigg, Stanislaw M. Ulam, and Samuel Yates. Immediate Source of Acquisition note Gift of Martin Gardner, 2002. Information about Access This collection is open for research. Ownership & Copyright All requests to reproduce, publish, quote from, or otherwise use collection materials must be submitted in writing to the Head of Special Collections and University Archives, Stanford University Libraries, Stanford, California 94304-6064. Consent is given on behalf of Special Collections as the owner of the physical items and is not intended to include or imply permission from the copyright owner. -
Chaos and Fractals Mew Frontiers of Science
Hartmutlurgens DietmarSaupe Heinz-Otto Peitgen Chaos and Fractals Mew Frontiers of Science With 686 illustrations, 40 in color Contents Preface VII Authors XI Foreword 1 Mitchell J. Feigenbaum Introduction: Causality Principle, Deterministic Laws and Chaos 9 1 The Backbone of Fractals: Feedback and the Iterator 15 1.1 The Principle of Feedback 17 1.2 The Multiple Reduction Copy Machine 23 1.3 Basic Types of Feedback Processes 27 1.4 The Parable of the Parabola — Or: Don't Trust Your Computer 37 1.5 Chaos Wipes Out Every Computer ' 49 1.6 Program of the Chapter: Graphical Iteration 60 2 Classical Fractals and Self-Similarity 63 2.1 The Cantor Set 67 2.2 The Sierpinski Gasket and Carpet 78 2.3 The Pascal Triangle 82 2.4 The Koch Curve 89 2.5 Space-Filling Curves 94 2.6 Fractals and the Problem of Dimension 106 2.7 The Universality of the Sierpinski Carpet 112 2.8 Julia Sets 122 2.9 Pythagorean Trees 126 2.10 Program of the Chapter: Sierpinski Gasket by Binary Addresses • : . 132 3 Limits and Self-Similarity 135 3.1 Similarity and Scaling 138 3.2 Geometric Series and the Koch Curve 147 3.3 Corner the New from Several Sides: Pi and the Square Root of Two 153 3.4 Fractals as Solution of Equations 168 3.5 Program of the Chapter: The Koch Curve 179 XIV Table of Contents 4 Length, Area and Dimension: Measuring Complexity and Scaling Properties 183 4.1 Finite and Infinite Length of Spirals 185 4.2 Measuring Fractal Curves and Power Laws 192 4.3 Fractal Dimension 202 4.4 The Box-Counting Dimension 212 4.5 Borderline Fractals: Devil's Staircase -
The H Fractal Self-Similarity Dimension Calculation Of
Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online) An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2014/04/jls.htm 2014 Vol. 4 (S4), pp. 3231-3235/Saeid et al. Research Article THE H FRACTAL SELF-SIMILARITY DIMENSION CALCULATION OF PARDIS TECHNOLOGY PARK IN TEHRAN (IRAN) *Saeid Rahmatabadi, Shabnam Akbari Namdar and Maryam Singery Department of Architecture, College of Art and Architecture, Tabriz Branch, Islamic Azad University, Tabriz, Iran *Author for Correspondence ABSTRACT The main purpose of this research is to find fractal designs in designing Pardis Technology Park in Tehran(Iran). In fact, we are seeking fractals in designing this work that has three characteristics of self- similarity, formation through repetition and lack of correct dimension. In this project, from the view of the author the site has a fractal, which is H- form that it is used in the plan and the plan form. We find the self-similarity dimension of the H fractal in this project. Keywords: Fractal, Self-similarity Dimension, Pardis Technology Park in Tehran(Iran) INTRODUCTION Fractals are self-similar sets whose patterns are made out of littler scales duplicated of them, having self- similarity crosswise over scales. This implies that they rehash the patterns to an unendingly little scale. An example with a higher fractal measurement is more confounded or eccentric than the one with a lower measurement, and fills more space. In numerous commonsense applications, worldly and spatial examination is expected to portray and measure the shrouded request in complex patterns, fractal geometry is a proper instrument for exploring such unpredictability over numerous scales for common phenomena (Mandelbrot, 1982; Burrough, 1983).