DOCSLIB.ORG

Special Topics in Mathematics: Fractals Course

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Special Topics in Mathematics: Fractals Course Department of Mathematics MAT 580 — SPECIAL TOPICS IN MATHEMATICS: FRACTALS COURSE DESCRIPTION Through hands-on activities, computer experiments, visuals, readings, discussions, and lectures, this course will examine the concepts of iteration, fractals, and chaos, as well as the history and people involved in this fascinating field. The course will begin by looking at the basic concept of iteration: repeatedly applying some specified operation. Iteration leads naturally to, among other things, the geometric representation of a fractal, self-similarity, fractal dimension, and the notion of mathematical chaos. Students will learn concepts from classical mathematics and ways in which these concepts can be applied to model and explain real-world scenarios. Additionally, they will be shown that mathematics is a current, active, useful, and dynamic field of study. 3 c.h., 3 s.h. RATIONALE The concept of fractional dimension has been around for about 80 years, but the term “fractal” itself and the interest in them, both popular and scientific, date from the proliferation of microcomputers in the early 1980s. For example, using the self-similarity feature of fractals allows for efficient data compression in such computer applications as Microsoft’s Encarta encyclopedia and in video games. As a result of all this interest, fractals have begun appearing in junior and high school textbooks as a way of demonstrating to today’s younger students that mathematics is an active and dynamic field of study, one for which all the answers are not known, indeed one for which all the questions have not even been asked. Our current mathematics curriculum at KU, both undergraduate and graduate, is decidedly more classical in nature. Indeed, we only peripherally touch on the subject of fractals in a few of our undergraduate courses. Because the study of fractals is a very popular, cutting edge, experimental branch of mathematics, it is one to which our students, in particular those in our Master of Education in Secondary Education with Specialization in Mathematics program, should be exposed. This course satisfies one of the course requirements in AREA IV: Subject Specialization of the Master of Education in Secondary Education degree. COURSE OBJECTIVES Students will be able to: Define the concepts of fractals, self-similarity, and fractal dimension, and and apply them to various other parts of mathematics. Generate classic fractals such as the Cantor Set, the Sierpinski Gasket, and the Koch Snowflake. Compute the limiting perimeter and area enclosed by fractal sets. Encode images by simple transformations. Use technology to perform the iterations needed to generate such fractal sets as Julia and Mandelbrot sets. Use their proof skills in the area of analysis as it applies to fractals and dynamical systems. Model real-world systems of interest so they can be investigated by mathematical tools. Explain how and when dynamical systems can demonstrate chaotic behavior. 1 COURSE OUTLINE I. Feedback and the Iterator A. The Principle of Feedback B. The Multiple Reduction Copy Machine (MRCM) C. Basic Types of Feedback Processes II. Classical Fractals and Self-Similarity A. The Cantor Set B. The Sierpinski Gasket and Carpet C. The Pascal Triangle D. The Koch Curve E. Space-Filling Curves F. Fractals and the Problem of Dimension G. The Universality of the Sierpinski Carpet H. Julia Sets I. Pythagorean Trees III. Limits and Self-Similarity A. Similarity and Scaling B. Geometric Series and the Koch Curve C. Comer the New from Several Sides: Pi and the Square Root of Two D. Fractals as Solution of Equations E. Box Self-Similarity: Grasping the Limit IV. Length, Area and Dimension: Measuring Complexity and Scaling Properties A. Finite and Infinite Length of Spirals B. Measuring Fractal Curves and Power Laws C. Fractal Dimension D. The Box-Counting Dimension E. Borderline Fractals: Devil's Staircase and Peano Curve V. Encoding Images by Simple Transformations A. The Multiple Reduction Copy Machine Metaphor B. Composing Simple Transformations C. Classical Fractals by Iterated Function Systems (IFSs) D. Image Encoding by IFSs . E. Foundation of IFS: The Contraction Mapping Principle F. Choosing the Right Metric G. Composing Self-Similar Images . H. Breaking Self-Similarity and Self-Affinity, or, Networking with MRCMs 2 VI. The Chaos Game: How Randomness Creates Deterministic Shapes A. The Fortune Wheel Reduction Copy Machine B. Addresses: Analysis of the Chaos Game C. Tuning the Fortune Wheel D. Random Number Generator Pitfall E. Adaptive Cut Methods VII. Irregular Shapes: Randomness in Fractal Constructions A. Randomizing Deterministic Fractals B. Percolation: Fractals and Fires in Random Forests C. Random Fractals in a Laboratory Experiment D. Simulation of Brownian Motion E. Scaling Laws and Fractional Brownian Motion F. Fractal Landscapes INSTRUCTIONAL RESOURCES Alligood, K. T. (1997). Chaos: An introduction to dynamical systems. New York: Springer Verlag. Bannon, T. J. (1991). Fractals and transformations. Mathematics Teacher , 84 (3), 178-85. Barcellos, A. (1984). The fractal geometry of Mandelbrot. College Mathematics Journal, 15 (2), 98-114. Barnsley, M. F. (1988). Fractals everywhere. Boston: Academic Press. Barton, R. (1990). Chaos and fractals. Mathematics Teacher, 83 (7), 524-29. Briggs, J. (1992). Fractals: the patterns of chaos: A new aesthetic of art, science, and nature. New York: Simon & Schuster. Briggs, J., & Peat, F.D. (1989). Turbulent mirror. NY: Harper & Row. Bunde, A., & Havlin, S. (Eds.). (1994). Fractals in science: With a MS-DOS program diskette. Berlin ; New York: Springer-Verlag. Camp, D. R. (1995). Fractal geometry in the high school classroom. International Reviews on Mathematical Education, 27 (5), 143-52. Camp, D., Chiaverina, C., & Senior, T. (1999) Festive fractals. Physics Teacher, 37 (9), 532. Choate, J. (1999). Fractals: A tool kit of dynamics activities. Emeryville, CA: Key Curriculum Press. Cibes, M. (1990). The Sierpinski triangle: Deterministic versus random models. Mathematics Teacher, 83 (8), 617-21. Crilly, A.J., Earnshaw, R.A., & Jones, H. (Eds.). (1993). Applications of fractals and chaos: The shape of things. New York: Springer-Verlag. 3 Coes, L. (1993). Building fractal models with manipulatives. Mathematics Teacher, 86 (8), 646-51. Davis, D. M. (1993). The nature and power of mathematics. Princeton, N.J.: Princeton University Press. Devaney, R. L. (1992). A first course in chaotic dynamical systems: Theory and experiment. Reading, MA: Addison-Wesley. Devaney, R. L. (2000). The Mandelbrot and Julia sets: A tool kit of dynamics activities. Emeryville, CA: Key Curriculum Press. **Dewdney, A. K. (1991). Exploring fractals. Science Probe, 1, 49-57. Edgar, G. A. (Ed.). (1993). Classics on fractals. Reading, MA: Addison-Wesley. **Eglash, R. (1999). African fractals. New Brunswick, NJ: Rutgers University Press. **Falconer, K. J. (2003). Fractal geometry : Mathematical foundations and applications. Chichester, NY: Wiley. Falconer, K. J. (1997). Techniques in fractal geometry. Chichester, NY: Wiley. **Fauvel, J., Flood, R., Wilson, R., & Wilson, R.J. (2003). Music and mathematics: From Pythagoras to fractals. New York: Oxford University Press. Feder, J. (1988). Fractals. New York: Plenum Press. Fisher, Y. (Ed.). (1995). Fractal image compression: Theory and application. New York: Springer-Verlag. Forster, P. (1997). Using fractals to teach complex numbers with a constructivist approach. Australian Senior Mathematics Journal, 11 (2) 14-22. **Gary, W.F. (2000). The computational beauty of nature: Computer explorations of fractals, chaos, complex systems, and adaptation. Cambridge, MA: MIT Press. Gazalé, M. J. (1999). Gnomon: From pharaohs to fractals. Princeton, N.J.: Princeton University Press. **Gleick, J. (1988). Chaos: Making a new science. East Rutherford, NJ: Penguin. Gullberg, J. (1997). Mathematics:From the birth of numbers. New York: W.W. Norton. **Havil, J. (2003). Gamma: Exploring Euler's constant. Princeton, NJ: Princeton University Press. Iannaccone, P. M., & Khokha, M. (1996). Fractal geometry in biological systems: An analytical approach. Boca Raton, FL: CRC Press. Jurgens, H. (1990). The language of fractals. Scientific American, 263, 60-67. **Lapidus, M. L., & van Frankenhuysen, M. (2000). Fractal geometry and number theory. Boston: Birkhauser. **Lesmoir-Gordon, N., Rood, W., Edney, R., & Appignanesi, R. (2000). Introducing fractal geometry. Blue Ridge Summit, PA: Totem Books. 4 Liebovitch, L. S. (1998). Fractals and chaos simplified for the life sciences. New York: Oxford University Press. Mandelbrot, B. B. (1977). Fractals: Form, chance and dimension. San Francisco: W. H. Freeman. Mandelbrot, B. B. (1983). The fractal geometry of nature. New York: W.H. Freeman. **Mandelbrot, B. B. (2004). Fractals and chaos. New York: Springer-Verlag. McCauley, J. L. (1993). Chaos, dynamics, and fractals: An algorithmic approach to deterministic chaos. New York, NY: Cambridge University Press. McGuire, M. D. (1991). An eye for fractals: A graphic & photographic essay. Redwood City, CA.: Addison-Wesley. **Le Mehaute, A. (1991). Fractal geometries: Theory and applications. Boca Raton, FL: CRC Press. Milner, J., Hodgson, C., Moore, K., & Wheatley, V. (2002). Fractals with Word. Mathematics Teaching , 178, 23-26. Moon, F. C. (1992). Chaotic and fractal dynamics: An introduction for applied scientists and engineers. New York: Wiley. Nievergelt, Y. (1991). Fractals illustrate the mathematical way of thinking. College Mathematics
Load more

Recommended publications
  • Mapping Inter and Transdisciplinary Relationships in Architecture: a First Approach to a Dictionary Under Construction
    Athens Journal of Architecture XY Mapping Inter and Transdisciplinary Relationships in Architecture: A First Approach to a Dictionary under Construction By Clara Germana Gonçalves Maria João Soares† This paper serves as part of the groundwork for the creation of a transdisciplinary dictionary of architecture that will be the result of inter and transdisciplinary research. The body of dictionary entries will be determined through the mapping of interrelating disciplines and concepts that emerge during said research. The aim is to create a dictionary whose entries derive from the scope of architecture as a discipline, some that may be not well defined in architecture but have full meanings in other disciplines. Or, even, define a hybrid disciplinary scope. As a first approach, the main entries – architecture and music, architecture and mathematics, architecture, music and mathematics, architecture and cosmology, architecture and dance, and architecture and cinema – incorporate secondary entries – harmony, matter and sound, full/void, organism and notation – for on the one hand these secondary entries present themselves as independent subjects (thought resulting from the main) and on the other they present themselves as paradigmatic representative concepts of a given conceptual context. It would also be of interest to see which concepts are the basis of each discipline and also those which, while not forming the basis, are essential to a discipline’s operationality. The discussion is focused in the context of history, the varying disciplinary interpretations, the differing implications those disciplinary interpretations have in the context of architecture, and the differing conceptual interpretations. The entries also make reference to important authors and studies in each specific case.
    [Show full text]
  • Review Article Survey Report on Space Filling Curves
    International Journal of Modern Science and Technology Vol. 1, No. 8, November 2016. Page 264-268. http://www.ijmst.co/ ISSN: 2456-0235. Review Article Survey Report on Space Filling Curves R. Prethee, A. R. Rishivarman Department of Mathematics, Theivanai Ammal College for Women (Autonomous) Villupuram - 605 401. Tamilnadu, India. *Corresponding author’s e-mail: [email protected] Abstract Space-filling Curves have been extensively used as a mapping from the multi-dimensional space into the one-dimensional space. Space filling curve represent one of the oldest areas of fractal geometry. Mapping the multi-dimensional space into one-dimensional domain plays an important role in every application that involves multidimensional data. We describe the notion of space filling curves and describe some of the popularly used curves. There are numerous kinds of space filling curves. The difference between such curves is in their way of mapping to the one dimensional space. Selecting the appropriate curve for any application requires knowledge of the mapping scheme provided by each space filling curve. Space filling curves are the basis for scheduling has numerous advantages like scalability in terms of the number of scheduling parameters, ease of code development and maintenance. The present paper report on various space filling curves, classifications, and its applications. It elaborates the space filling curves and their applicability in scheduling, especially in transaction. Keywords: Space filling curve, Holder Continuity, Bi-Measure-Preserving Property, Transaction Scheduling. Introduction these other curves, sometimes space-filling In mathematical analysis, a space-filling curves are still referred to as Peano curves. curve is a curve whose range contains the entire Mathematical tools 2-dimensional unit square or more generally an The Euclidean Vector Norm n-dimensional unit hypercube.
    [Show full text]
  • Fractal-Based Magnetic Resonance Imaging Coils for 3T Xenon Imaging Fractal-Based Magnetic Resonance Imaging Coils for 3T Xenon Imaging
    Fractal-based magnetic resonance imaging coils for 3T Xenon imaging Fractal-based magnetic resonance imaging coils for 3T Xenon imaging By Jimmy Nguyen A Thesis Submitted to the School of Graduate Studies in thePartial Fulfillment of the Requirements for the Degree Master of Applied Science McMaster University © Copyright by Jimmy Nguyen 10 July 2020 McMaster University Master of Applied Science (2020) Hamilton, Ontario (Department of Electrical & Computer Engineering) TITLE: Fractal-based magnetic resonance imaging coils for 3T Xenon imaging AUTHOR: Jimmy Nguyen, B.Eng., (McMaster University) SUPERVISOR: Dr. Michael D. Noseworthy NUMBER OF PAGES: ix, 77 ii Abstract Traditional 1H lung imaging using MRI faces numerous challenges and difficulties due to low proton density and air-tissue susceptibility artifacts. New imaging techniques using inhaled xenon gas can overcome these challenges at the cost of lower signal to noise ratio. The signal to noise ratio determines reconstructed image quality andis an essential parameter in ensuring reliable results in MR imaging. The traditional RF surface coils used in MR imaging exhibit an inhomogeneous field, leading to reduced image quality. For the last few decades, fractal-shaped antennas have been used to optimize the performance of antennas for radiofrequency systems. Although widely used in radiofrequency identification systems, mobile phones, and other applications, fractal designs have yet to be fully researched in the MRI application space. The use of fractal geometries for RF coils may prove to be fruitful and thus prompts an investiga- tion as the main goal of this thesis. Preliminary simulation results and experimental validation results show that RF coils created using the Gosper and pentaflake offer improved signal to noise ratio and exhibit a more homogeneous field than that ofa traditional circular surface coil.
    [Show full text]
  • MATHEMATICS EXTENDED ESSAY Mathematics in Music
    TED ANKARA COLLEGE FOUNDATION PRIVATE HIGH SCHOOL THE INTERNATIONAL BACCALAUREATE PROGRAMME MATHEMATICS EXTENDED ESSAY Mathematics In Music Candidate: Su Güvenir Supervisor: Mehmet Emin ÖZER Candidate Number: 001129‐047 Word Count: 3686 words Research Question: How mathematics and music are related to each other in the way of traditional and patternist perspective? Su Güvenir, 001129 ‐ 047 Abstract Links between music and mathematics is always been an important research topic ever. Serious attempts have been made to identify these links. Researches showed that relationships are much more stronger than it has been anticipated. This paper identifies these relationships between music and various fields of mathematics. In addition to these traditional mathematical investigations to the field of music, a new approach is presented: Patternist perspective. This new perspective yields new important insights into music theory. These insights are explained in a detailed way unraveling some interesting patterns in the chords and scales. The patterns in the underlying the chords are explained with the help of an abstract device named chord wheel which clearly helps to visualize the most essential relationships in the chord theory. After that, connections of music with some fields in mathematics, such as fibonacci numbers and set theory, are presented. Finally concluding that music is an abstraction of mathematics. Word Count: 154 words. 2 Su Güvenir, 001129 ‐ 047 TABLE OF CONTENTS ABSTRACT………………………………………………………………………………………………………..2 CONTENTS PAGE……………………………………………………………………………………………...3
    [Show full text]
  • Harmonious Hilbert Curves and Other Extradimensional Space-Filling Curves
    Harmonious Hilbert curves and other extradimensional space-filling curves∗ Herman Haverkorty November 2, 2012 Abstract This paper introduces a new way of generalizing Hilbert's two-dimensional space-filling curve to arbitrary dimensions. The new curves, called harmonious Hilbert curves, have the unique property that for any d0 < d, the d-dimensional curve is compatible with the d0-dimensional curve with respect to the order in which the curves visit the points of any d0-dimensional axis-parallel space that contains the origin. Similar generalizations to arbitrary dimensions are described for several variants of Peano's curve (the original Peano curve, the coil curve, the half-coil curve, and the Meurthe curve). The d-dimensional harmonious Hilbert curves and the Meurthe curves have neutral orientation: as compared to the curve as a whole, arbitrary pieces of the curve have each of d! possible rotations with equal probability. Thus one could say these curves are `statistically invariant' under rotation|unlike the Peano curves, the coil curves, the half-coil curves, and the familiar generalization of Hilbert curves by Butz and Moore. In addition, prompted by an application in the construction of R-trees, this paper shows how to construct a 2d-dimensional generalized Hilbert or Peano curve that traverses the points of a certain d-dimensional diagonally placed subspace in the order of a given d-dimensional generalized Hilbert or Peano curve. Pseudocode is provided for comparison operators based on the curves presented in this paper. 1 Introduction Space-filling curves A space-filling curve in d dimensions is a continuous, surjective mapping from R to Rd.
    [Show full text]
  • An Exploration of the Relationship Between Mathematics and Music
    An Exploration of the Relationship between Mathematics and Music Shah, Saloni 2010 MIMS EPrint: 2010.103 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 An Exploration of ! Relation"ip Between Ma#ematics and Music MATH30000, 3rd Year Project Saloni Shah, ID 7177223 University of Manchester May 2010 Project Supervisor: Professor Roger Plymen ! 1 TABLE OF CONTENTS Preface! 3 1.0 Music and Mathematics: An Introduction to their Relationship! 6 2.0 Historical Connections Between Mathematics and Music! 9 2.1 Music Theorists and Mathematicians: Are they one in the same?! 9 2.2 Why are mathematicians so fascinated by music theory?! 15 3.0 The Mathematics of Music! 19 3.1 Pythagoras and the Theory of Music Intervals! 19 3.2 The Move Away From Pythagorean Scales! 29 3.3 Rameau Adds to the Discovery of Pythagoras! 32 3.4 Music and Fibonacci! 36 3.5 Circle of Fifths! 42 4.0 Messiaen: The Mathematics of his Musical Language! 45 4.1 Modes of Limited Transposition! 51 4.2 Non-retrogradable Rhythms! 58 5.0 Religious Symbolism and Mathematics in Music! 64 5.1 Numbers are God"s Tools! 65 5.2 Religious Symbolism and Numbers in Bach"s Music! 67 5.3 Messiaen"s Use of Mathematical Ideas to Convey Religious Ones! 73 6.0 Musical Mathematics: The Artistic Aspect of Mathematics! 76 6.1 Mathematics as Art! 78 6.2 Mathematical Periods! 81 6.3 Mathematics Periods vs.
    [Show full text]
  • Mathematics As “Gate-Keeper” (?): Three Theoretical Perspectives That Aim Toward Empowering All Children with a Key to the Gate DAVID W
    ____ THE _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 14 Number 1 Spring 2004 MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA Editorial Staff A Note from the Editor Editor Dear TME readers, Holly Garrett Anthony Along with the editorial team, I present the first of two issues to be produced during my brief tenure as editor of Volume 14 of The Mathematics Educator. This issue showcases the work of both Associate Editors veteran and budding scholars in mathematics education. The articles range in topic and thus invite all Ginger Rhodes those vested in mathematics education to read on. Margaret Sloan Both David Stinson and Amy Hackenberg direct our attention toward equity and social justice in Erik Tillema mathematics education. Stinson discusses the “gatekeeping” status of mathematics, offers theoretical perspectives he believes can change this, and motivates mathematics educators at all levels to rethink Publication their roles in empowering students. Hackenberg’s review of Burton’s edited book, Which Way Social Stephen Bismarck Justice in Mathematics Education? is both critical and engaging. She artfully draws connections across Laurel Bleich chapters and applauds the picture of social justice painted by the diversity of voices therein. Dennis Hembree Two invited pieces, one by Chandra Orrill and the other by Sybilla Beckmann, ask mathematics Advisors educators to step outside themselves and reexamine features of PhD programs and elementary Denise S. Mewborn textbooks. Orrill’s title question invites mathematics educators to consider what we value in classroom Nicholas Oppong teaching, how we engage in and write about research on or with teachers, and what features of a PhD program can inform teacher education.
    [Show full text]
  • Review On: Fractal Antenna Design Geometries and Its Applications
    www.ijecs.in International Journal Of Engineering And Computer Science ISSN:2319-7242 Volume - 3 Issue -9 September, 2014 Page No. 8270-8275 Review On: Fractal Antenna Design Geometries and Its Applications Ankita Tiwari1, Dr. Munish Rattan2, Isha Gupta3 1GNDEC University, Department of Electronics and Communication Engineering, Gill Road, Ludhiana 141001, India [email protected] 2GNDEC University, Department of Electronics and Communication Engineering, Gill Road, Ludhiana 141001, India [email protected] 3GNDEC University, Department of Electronics and Communication Engineering, Gill Road, Ludhiana 141001, India [email protected] Abstract: In this review paper, we provide a comprehensive review of developments in the field of fractal antenna engineering. First we give brief introduction about fractal antennas and then proceed with its design geometries along with its applications in different fields. It will be shown how to quantify the space filling abilities of fractal geometries, and how this correlates with miniaturization of fractal antennas. Keywords – Fractals, self -similar, space filling, multiband 1. Introduction Modern telecommunication systems require antennas with irrespective of various extremely irregular curves or shapes wider bandwidths and smaller dimensions as compared to the that repeat themselves at any scale on which they are conventional antennas. This was beginning of antenna research examined. in various directions; use of fractal shaped antenna elements was one of them. Some of these geometries have been The term “Fractal” means linguistically “broken” or particularly useful in reducing the size of the antenna, while “fractured” from the Latin “fractus.” The term was coined by others exhibit multi-band characteristics. Several antenna Benoit Mandelbrot, a French mathematician about 20 years configurations based on fractal geometries have been reported ago in his book “The fractal geometry of Nature” [5].
    [Show full text]
  • Spatial Accessibility to Amenities in Fractal and Non Fractal Urban Patterns Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser
    Spatial accessibility to amenities in fractal and non fractal urban patterns Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser To cite this version: Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser. Spatial accessibility to amenities in fractal and non fractal urban patterns. Environment and Planning B: Planning and Design, SAGE Publications, 2012, 39 (5), pp.801-819. 10.1068/b37132. hal-00804263 HAL Id: hal-00804263 https://hal.archives-ouvertes.fr/hal-00804263 Submitted on 14 Jun 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. TANNIER C., VUIDEL G., HOUOT H., FRANKHAUSER P. (2012), Spatial accessibility to amenities in fractal and non fractal urban patterns, Environment and Planning B: Planning and Design, vol. 39, n°5, pp. 801-819. EPB 137-132: Spatial accessibility to amenities in fractal and non fractal urban patterns Cécile TANNIER* ([email protected]) - corresponding author Gilles VUIDEL* ([email protected]) Hélène HOUOT* ([email protected]) Pierre FRANKHAUSER* ([email protected]) * ThéMA, CNRS - University of Franche-Comté 32 rue Mégevand F-25 030 Besançon Cedex, France Tel: +33 381 66 54 81 Fax: +33 381 66 53 55 1 Spatial accessibility to amenities in fractal and non fractal urban patterns Abstract One of the challenges of urban planning and design is to come up with an optimal urban form that meets all of the environmental, social and economic expectations of sustainable urban development.
    [Show full text]
  • Redalyc.Self-Similarity of Space Filling Curves
    Ingeniería y Competitividad ISSN: 0123-3033 [email protected] Universidad del Valle Colombia Cardona, Luis F.; Múnera, Luis E. Self-Similarity of Space Filling Curves Ingeniería y Competitividad, vol. 18, núm. 2, 2016, pp. 113-124 Universidad del Valle Cali, Colombia Available in: http://www.redalyc.org/articulo.oa?id=291346311010 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 Ingeniería y Competitividad, Volumen 18, No. 2, p. 113 - 124 (2016) COMPUTATIONAL SCIENCE AND ENGINEERING Self-Similarity of Space Filling Curves INGENIERÍA DE SISTEMAS Y COMPUTACIÓN Auto-similaridad de las Space Filling Curves Luis F. Cardona*, Luis E. Múnera** *Industrial Engineering, University of Louisville. KY, USA. ** ICT Department, School of Engineering, Department of Information and Telecommunication Technologies, Faculty of Engineering, Universidad Icesi. Cali, Colombia. [email protected]*, [email protected]** (Recibido: Noviembre 04 de 2015 – Aceptado: Abril 05 de 2016) Abstract We define exact self-similarity of Space Filling Curves on the plane. For that purpose, we adapt the general definition of exact self-similarity on sets, a typical property of fractals, to the specific characteristics of discrete approximations of Space Filling Curves. We also develop an algorithm to test exact self- similarity of discrete approximations of Space Filling Curves on the plane. In addition, we use our algorithm to determine exact self-similarity of discrete approximations of four of the most representative Space Filling Curves.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]