Fractals.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Fractals Self Similarity and Fractal Geometry presented by Pauline Jepp 601.73 Biological Computing Overview History Initial Monsters Details Fractals in Nature Brownian Motion L-systems Fractals defined by linear algebra operators Non-linear fractals History Euclid's 5 postulates: 1. To draw a straight line from any point to any other. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to each other. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles. History Euclid ~ "formless" patterns Mandlebrot's Fractals "Pathological" "gallery of monsters" In 1875: Continuous non-differentiable functions, ie no tangent La Femme Au Miroir 1920 Leger, Fernand Initial Monsters 1878 Cantor's set 1890 Peano's space filling curves Initial Monsters 1906 Koch curve 1960 Sierpinski's triangle Details Fractals : are self similar fractal dimension A square may be broken into N^2 self-similar pieces, each with magnification factor N Details Effective dimension Mandlebrot: " ... a notion that should not be defined precisely. It is an intuitive and potent throwback to the Pythagoreans' archaic Greek geometry" How long is the coast of Britain? Steinhaus 1954, Richardson 1961 Brownian Motion Robert Brown 1827 Jean Perrin 1906 Diffusion-limited aggregation L-Systems and Fractal Growth Packing efficiency Axiom & production rules Axiom: B Rules: B->F[-B]+B F->FF B F[-B]+B FF[-F[-B]+B]+F[-B]+B L-Systems and Fractal Growth Turtle graphics Seymour Papert L-Systems and Fractal Growth L-Systems and Fractal Growth L-Systems and Fractal Growth Affine Transformation Fractals "It has a miniature version of itself embedded inside it, but the smaller version is slightly rotated." Transofrmations: translation, scale, reflection rotation Affine Transformation Fractals Michael Barnsley Multiple Reduction Copy Machine Algorithm (MRCM) Affine Transformation Fractals Multiple Reduction Copy Machine Algorithm (MRCM) Affine Transformation Fractals Iterated Functional Systems (IFS) Affine Transformation Fractals Iterated Functional Systems (IFS) Affine Transformation Fractals Iterated Functional Systems (IFS) Affine Transformation Fractals Iterated Functional Systems (IFS) The Mandelbrot & Julia sets Iterative Dynamical Systems The Mandelbrot & Julia sets For each number, c, in a subset of the complex plane Set x0 = 0 For t = 1 to tmax 2 Compute xt = x t + c If t< tmax, then colour point c white If t = tmax, then colour point c black The Mandelbrot & Julia sets The Mandelbrot & Julia sets M-Set and computability cardoid: x = 1/4(2 cos t - cos 2t ) y = 1/4(2 sint t - sin 2 t ) The Mandelbrot & Julia sets The M-Set as the Master Julia set. Set c to some constant complex value For each number, x0 in a subset of the complex plane For t = 1 to tmax 2 Compute xt = xt + c If |xt| > 2 then break out of loop If t < tmax then colour point c white It t = tmax thencolour point c black The Mandelbrot & Julia sets The Mandelbrot & Julia sets.

Copy Link

Recommended publications

Ergodic Theory, Fractal Tops and Colour Stealing 1
ERGODIC THEORY, FRACTAL TOPS AND COLOUR STEALING MICHAEL BARNSLEY Abstract. A new structure that may be associated with IFS and superIFS is described. In computer graphics applications this structure can be rendered using a new algorithm, called the “colour stealing”. 1. Ergodic Theory and Fractal Tops The goal of this lecture is to describe informally some recent realizations and work in progress concerning IFS theory with application to the geometric modelling and assignment of colours to IFS fractals and superfractals. The results will be described in the simplest setting of a single IFS with probabilities, but many generalizations are possible, most notably to superfractals. Let the iterated function system (IFS) be denoted (1.1) X; f1, ..., fN ; p1,...,pN . { } This consists of a ﬁnite set of contraction mappings (1.2) fn : X X,n=1, 2, ..., N → acting on the compact metric space (1.3) (X,d) with metric d so that for some (1.4) 0 l<1 ≤ we have (1.5) d(fn(x),fn(y)) l d(x, y) ≤ · for all x, y X.Thepn’s are strictly positive probabilities with ∈ (1.6) pn =1. n X The probability pn is associated with the map fn. We begin by reviewing the two standard structures (one and two)thatare associated with the IFS 1.1, namely its set attractor and its measure attractor, with emphasis on the Collage Property, described below. This property is of particular interest for geometrical modelling and computer graphics applications because it is the key to designing IFSs with attractor structures that model given inputs.
[Show full text]
Fractal Interpolation
University of Central Florida STARS Electronic Theses and Dissertations, 2004-2019 2008 Fractal Interpolation Gayatri Ramesh University of Central Florida Part of the Mathematics Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Ramesh, Gayatri, "Fractal Interpolation" (2008). Electronic Theses and Dissertations, 2004-2019. 3687. https://stars.library.ucf.edu/etd/3687 FRACTAL INTERPOLATION by GAYATRI RAMESH B.S. University of Tennessee at Martin, 2006 A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics in the College of Sciences at the University of Central Florida Orlando, Florida Fall Term 2008 ©2008 Gayatri Ramesh ii ABSTRACT This thesis is devoted to a study about Fractals and Fractal Polynomial Interpolation. Fractal Interpolation is a great topic with many interesting applications, some of which are used in everyday lives such as television, camera, and radio. The thesis is comprised of eight chapters. Chapter one contains a brief introduction and a historical account of fractals. Chapter two is about polynomial interpolation processes such as Newton’s, Hermite, and Lagrange. Chapter three focuses on iterated function systems. In this chapter I report results contained in Barnsley’s paper, Fractal Functions and Interpolation. I also mention results on iterated function system for fractal polynomial interpolation.
[Show full text]
Current Practices in Quantitative Literacy © 2006 by the Mathematical Association of America (Incorporated)
Current Practices in Quantitative Literacy © 2006 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2005937262 Print edition ISBN: 978-0-88385-180-7 Electronic edition ISBN: 978-0-88385-978-0 Printed in the United States of America Current Printing (last digit): 10 9 8 7 6 5 4 3 2 1 Current Practices in Quantitative Literacy edited by Rick Gillman Valparaiso University Published and Distributed by The Mathematical Association of America The MAA Notes Series, started in 1982, addresses a broad range of topics and themes of interest to all who are in- volved with undergraduate mathematics. The volumes in this series are readable, informative, and useful, and help the mathematical community keep up with developments of importance to mathematics. Council on Publications Roger Nelsen, Chair Notes Editorial Board Sr. Barbara E. Reynolds, Editor Stephen B Maurer, Editor-Elect Paul E. Fishback, Associate Editor Jack Bookman Annalisa Crannell Rosalie Dance William E. Fenton Michael K. May Mark Parker Susan F. Pustejovsky Sharon C. Ross David J. Sprows Andrius Tamulis MAA Notes 14. Mathematical Writing, by Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts. 16. Using Writing to Teach Mathematics, Andrew Sterrett, Editor. 17. Priming the Calculus Pump: Innovations and Resources, Committee on Calculus Reform and the First Two Years, a subcomit- tee of the Committee on the Undergraduate Program in Mathematics, Thomas W. Tucker, Editor. 18. Models for Undergraduate Research in Mathematics, Lester Senechal, Editor. 19. Visualization in Teaching and Learning Mathematics, Committee on Computers in Mathematics Education, Steve Cunningham and Walter S.
[Show full text]
Fractal Image Compression, AK Peters, Wellesley, 1992
FRACTAL STRUCTURES: IMAGE ENCODING AND COMPRESSION TECHNIQUES V. Drakopoulos Department of Computer Science and Biomedical Informatics 2 Applications • Biology • Botany • Chemistry • Computer Science (Graphics, Vision, Image Processing and Synthesis) • Geology • Mathematics • Medicine • Physics 3 Bibliography • Barnsley M. F., Fractals everywhere, 2nd ed., Academic Press Professional, San Diego, CA, 1993. • Barnsley M. F., SuperFractals, Cambridge University Press, New York, 2006. • Barnsley M. F. and Anson, L. F., The fractal transform, Jones and Bartlett Publishers, Inc, 1993. • Barnsley M. F. and Hurd L. P., Fractal image compression, AK Peters, Wellesley, 1992. • Barnsley M. F., Saupe D. and Vrscay E. R. (eds.), Fractals in multimedia, Springer- Verlag, New York, 2002. • Falconer K. J., Fractal geometry: Mathematical foundations and applications, Wiley, Chichester, 1990. • Fisher Y., Fractal image compression (ed.), Springer-Verlag, New York, 1995. • Hoggar S. G., Mathematics for computer graphics, Cambridge University Press, Cambridge, 1992. • Lu N., Fractal imaging, Academic Press, San Diego, CA, 1997. • Mandelbrot B. B., Fractals: Form, chance and dimension, W. H. Freeman, San Francisco, 1977. • Mandelbrot B. B., The fractal geometry of nature, W. H. Freeman, New York, 1982. • Massopust P. R., Fractal functions, fractal surfaces and wavelets, Academic Press, San Diego, CA, 1994. • Nikiel S., Iterated function systems for real-time image synthesis, Springer-Verlag, London, 2007. • Peitgen H.--O. and Saupe D. (eds.), The science
[Show full text]
Math Morphing Proximate and Evolutionary Mechanisms
Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 2009 Volume V: Evolutionary Medicine Math Morphing Proximate and Evolutionary Mechanisms Curriculum Unit 09.05.09 by Kenneth William Spinka Introduction Background Essential Questions Lesson Plans Website Student Resources Glossary Of Terms Bibliography Appendix Introduction An important theoretical development was Nikolaas Tinbergen's distinction made originally in ethology between evolutionary and proximate mechanisms; Randolph M. Nesse and George C. Williams summarize its relevance to medicine: All biological traits need two kinds of explanation: proximate and evolutionary. The proximate explanation for a disease describes what is wrong in the bodily mechanism of individuals affected Curriculum Unit 09.05.09 1 of 27 by it. An evolutionary explanation is completely different. Instead of explaining why people are different, it explains why we are all the same in ways that leave us vulnerable to disease. Why do we all have wisdom teeth, an appendix, and cells that if triggered can rampantly multiply out of control? [1] A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Beno?t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. http://www.kwsi.com/ynhti2009/image01.html A fractal often has the following features: 1. It has a fine structure at arbitrarily small scales.
[Show full text]
Chaos Theory: the Essential for Military Applications
U.S. Naval War College U.S. Naval War College Digital Commons Newport Papers Special Collections 10-1996 Chaos Theory: The Essential for Military Applications James E. Glenn Follow this and additional works at: https://digital-commons.usnwc.edu/usnwc-newport-papers Recommended Citation Glenn, James E., "Chaos Theory: The Essential for Military Applications" (1996). Newport Papers. 10. https://digital-commons.usnwc.edu/usnwc-newport-papers/10 This Book is brought to you for free and open access by the Special Collections at U.S. Naval War College Digital Commons. It has been accepted for inclusion in Newport Papers by an authorized administrator of U.S. Naval War College Digital Commons. For more information, please contact [email protected]. The Newport Papers Tenth in the Series CHAOS ,J '.' 'l.I!I\'lt!' J.. ,\t, ,,1>.., Glenn E. James Major, U.S. Air Force NAVAL WAR COLLEGE Chaos Theory Naval War College Newport, Rhode Island Center for Naval Warfare Studies Newport Paper Number Ten October 1996 The Newport Papers are extended research projects that the editor, the Dean of Naval Warfare Studies, and the President of the Naval War CoJIege consider of particular in terest to policy makers, scholars, and analysts. Papers are drawn generally from manuscripts not scheduled for publication either as articles in the Naval War CollegeReview or as books from the Naval War College Press but that nonetheless merit extensive distribution. Candidates are considered by an edito rial board under the auspices of the Dean of Naval Warfare Studies. The views expressed in The Newport Papers are those of the authors and not necessarily those of the Naval War College or the Department of the Navy.
[Show full text]
Organizing the City: Morphology and Dynamics Phenomenology of a Spatial Hierarchy
Die approbierte Originalversion dieser Dissertation ist an der Hauptbibliothek der Technischen Universität Wien aufgestellt (http://www.ub.tuwien.ac.at). The approved original version of this thesis is available at the main library of the Vienna University of Technology (http://www.ub.tuwien.ac.at/englweb/). DISSERTATION ORGANIZING THE CITY: MORPHOLOGY AND DYNAMICS PHENOMENOLOGY OF A SPATIAL HIERARCHY ausgeführt zum Zwecke der Erlangung des akademischen Grades eines Doktors der technischen Wissenschaften Thesis submitted for the degree of Doctor technicae Author: Dipl.-Ing. Claudia Czerkauer Matr.# 9625752, E 086 600 Rotenhofgasse 49, 1100 Vienna Austria Under the supervision of: Univ. Prof. Dipl. Ing. Dr. Georg Franck-Oberaspach IEMAR, Department for Digital Architecture and Urban Planning Vienna University of Technology, Austria Univ. Prof. Dr. Nikos A. Salingaros, MA Department of Mathematics, University of Texas at San Antonio, USA Submitted at: Vienna University of Technology Faculty for Architecture & Urban and Regional Planning Austria Vienna, November 2007_________________________ ORGANIZING THE CITY: MORPHOLOGY AND DYNAMICS PHENOMENOLOGY OF A SPATIAL HIERARCHY for Christoph Acknowledgements I should like to thank many people who have contributed to the work described here: Firstly, my supervisors Prof. Dr. Georg Franck-Oberaspach and Prof. Dr. Nikos Salingaros for their motivation and inspiration. In particular, Prof. Dr. Dr. Pierre Frankhauser, Université de Franche-Comté, France, and Prof. Dr. Bill Hillier, University of London & Space Syntax Ltd., London, where I had the opportunity to join their laboratories and work on my research with a lot of support from their teams. Also, to Dipl. Ing. Andreas Nuß, MA 18 Vienna, and Roswitha Lacina, Department of Urban Planning, Vienna University of Technology, for their expertise.
[Show full text]
Self-Similar Polygonal Tiling
Mathematical Assoc. of America American Mathematical Monthly 121:1 November 11, 2016 10:48 a.m. SSPfinal.tex page 1 Self-Similar Polygonal Tiling Michael Barnsley and Andrew Vince Abstract. The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting, and to provide a novel method for the construction of such polygonal tilings. 1. INTRODUCTION. Our goal is to lure the reader into the theory underlying the figures scattered throughout this paper. The individual polygonal tiles in each of these tilings are pairwise similar, and there are only finitely many up to congruence. Each tiling is self-similar. None of the tilings are periodic, yet each is quasiperiodic. These concepts, self-similarity and quasiperiodicity, are defined in Section 3 and are dis- cussed throughout the paper. Each tiling is constructed by the same method from a single self-similar polygon. Figure 1. A self-similar polygonal tiling of order 2. For the tiling T of the plane, a part of which is shown in Figure 1, there are two sim- ilar tile shapes,p the ratio of the sides of the larger quadrilateral to the smaller quadrilat- eral being 3 : 1. In the tiling of the entire plane, the part shown in the figure appears “everywhere,” the phenomenon known as quasiperiodicity or repetitivity. The tiling is self-similar in that there exists a similarity transformation φ of the plane such that, for each tile t 2 T , the “blown up” tile φ(t) = fφ(x): x 2 tg is the disjoint union of the original tiles in T .
[Show full text]
Glimpses of Benoît B. Mandelbrot (1924–2010) Edited by Michael F
Glimpses of Benoît B. Mandelbrot (1924–2010) Edited by Michael F. Barnsley and Michael Frame Two roads diverged in a wood, and I — I took the one less travelled by, And that has made all the diﬀerence. (Robert Frost, “The Road Not Taken”) Michael F. Barnsley Introduction Benoît B. Mandelbrot died in Cambridge, Mas- sachusetts, on Thursday, 14 October 2010. He was eighty-ﬁve years old and Sterling Professor Emeri- tus of Mathematical Sciences at Yale University. He was also IBM Fellow Emeritus (physics) at the IBM T. J. Watson Research Center. He was a great and rare mathematician and scientist. He changed the way that many of us see, describe and model, math- ematically and geometrically, the world around us. He moved between disciplines and university de- Figure 1. Benoît Mandelbrot next to John partments, from geology to physics, to computer Robinson’s sculpture Intuition outside the Isaac science, to economics and engineering, talking Newton Institute, Cambridge, during the excitedly, sometimes obscurely, strangely vain, Mathematics and Applications of Fractals about all manner of things, theorizing, speculat- Program in 1999. (Photo: Findlay Kember/Isaac ing, and often in recent years, to the annoyance Newton Institute.) of others, pointing out how he had earlier done work of a related nature to whatever it was that someone was explaining, bobbing up and down to Looking back, Benoît saw his life as a rough path. interrupt, to explain this or that. He was an un- In [7] he recounted how his father escaped from forgettable, extraordinary person of great warmth Poland and the Nazis with a group of others and, at who was also vulnerable and defensive.
[Show full text]
Redalyc.FRACTAL ECONOMY FUNCTIONS: COGNITIVE AND
Investigación Administrativa ISSN: 1870-6614 [email protected] Escuela Superior de Comercio y Administración, Unidad Santo Tomás México Barnsley, Michael; Ramos, María; Villasante, Sebastian FRACTAL ECONOMY FUNCTIONS: COGNITIVE AND PARTICIPATION Investigación Administrativa, núm. 110, julio-diciembre, 2012, pp. 49-56 Escuela Superior de Comercio y Administración, Unidad Santo Tomás Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=456045338004 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative FUNCIONES DE LA ECONOMÍA FRACTAL: COGNICIÓN Y PARTICIPACIÓN FRACTAL ECONOMY FUNCTIONS: COGNITIVE AND PARTICIPATION Michael Barnsley (1) María Ramos (2) Sebastian Villasante (3) ABSTRACT In this paper we present fractal modeling that will help us understand and analyze the prospects of reaching the stations listed its shares in the financial market and the correlation of the latter with the company, including relaxation of the ranges we purchase and sales functions sine and cosine to serve as basis for investment circles via fractal Fibonacci series is non-parametric statistics with golden mean. Key words: Fractal, iteration investment circle, alternating parameter, smoothing. RESUMEN En este trabajo se presenta el modelado fractal que nos ayudará a comprender y analizar las 49 perspectivas de alcanzar prospecciones de las acciones en el mercado financiero y la correlación de ésta con la empresa, incluyendo la relajación de los rangos de nosotros compra y venta de seno y coseno funciones para servir como base para la inversión a través de los círculos fractal serie de Fibonacci es estadística no paramétrica con media dorada.
[Show full text]
Dimensions of Self-Similar Fractals
DIMENSIONS OF SELF-SIMILAR FRACTALS BY MELISSA GLASS A Thesis Submitted to the Graduate Faculty of WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES in Partial Fulﬁllment of the Requirements for the Degree of MASTER OF ARTS Mathematics May 2011 Winston-Salem, North Carolina Approved By: Sarah Raynor, Ph.D., Advisor James Kuzmanovich, Ph.D., Chair Jason Parsley, Ph.D. Acknowledgments First and foremost, I would like to thank God for making me who I am, for His guidance, and for giving me the strength and ability to write this thesis. Second, I would like to thank my husband Andy for all his love, support and patience even on the toughest days. I appreciate the countless dinners he has made me while I worked on this thesis. Next, I would like to thank Dr. Raynor for all her time, knowledge, and guidance throughout my time at Wake Forest. I would also like to thank her for pushing me to do my best. A special thank you goes out to Dr. Parsley for putting up with me in class! I also appreciate him serving as one of my thesis committee members. I also appreciate all the fun conversations with Dr. Kuzmanovich whether it was about mathematics or mushrooms! Also, I would like to thank him for serving as one of my thesis committee members. A thank you is also in order for all the professors at Wake Forest who have taught me mathematics. ii Table of Contents Acknowledgments . ii List of Figures. v Abstract . vi Chapter 1 Introduction . 1 1.1 Motivation .
[Show full text]
Fractal Geometry: History and Theory
Smith 1 Geri Smith MATH H324 College Geometry Dr. Kent Honors Research Paper April 26 th , 2011 Fractal Geometry: History and Theory Classical Euclidean geometry cannot accurately represent the natural world; fractal geometry is the geometry of nature. Fractal geometry can be described as an extension of Euclidean geometry and can create concrete models of the various physical structures within nature. In short, fractal geometry and fractals are characterized by self-similarity and recursion, which entails scaling patterns, patterns within patterns, and symmetry across every scale. Benoit Mandelbrot, the “father” of fractal geometry, coined the term “fractal,” in the1970s, from the Latin “Fractus” (broken), to describe these infinitely complex scaling shapes. Yet, the basic ideas behind fractals were explored as far back as the seventeenth century; however, the oldest fractal is considered to be the nineteenth century's Cantor set. A variety of mathematical curiosities, “pathological monsters,” like the Cantor set, the Julia set, and the Peano space-filling curve, upset 19 th century standards and confused mathematicians, who believed this demonstrated the ability to push mathematics out of the realm of the natural world. It was not until fractal geometry was developed in the 1970s that what past mathematicians thought to be unnatural was shown to be truly representative of natural phenomena. In the late 1970s and throughout the 1980s, fractal geometry captured world-wide interest, even among non-mathematicians (probably due to the fact fractals make for pretty pictures), and was a popular topic- conference sprung up and Mandelbrot's treatise on fractal geometry, The Fractal Geometry of Nature, was well-read even outside of the math community.
[Show full text]