Louisiana State University LSU Digital Commons LSU Master's Theses Graduate School 2016 Fractal Shapes Generated by Iterated Function Systems Mary Catherine McKinley Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_theses Part of the Applied Mathematics Commons Recommended Citation McKinley, Mary Catherine, "Fractal Shapes Generated by Iterated Function Systems" (2016). LSU Master's Theses. 4571. https://digitalcommons.lsu.edu/gradschool_theses/4571 This Thesis is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Master's Theses by an authorized graduate school editor of LSU Digital Commons. For more information, please contact [email protected]. FRACTAL SHAPES GENERATED BY ITERATED FUNCTION SYSTEMS A Thesis Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Masters of Science in Mathematics by Mary Catherine McKinley BS, Spring Hill College, 2015 December 2016 Acknowledgments The author would like to thank Dr. Frank Neubrander for his support and guidance on this project; Dr. James Madden for his help with the Mathematica programs; Dr. Oliver Dasbach and Dr. James Oxley for their unending support and encouragement; James Frye for listening and believing in me; and most of all my family for being taught the beauty of fractals during this project and never complaining about it. ii Table of Contents Acknowledgments . ii List of Tables . iv List of Figures . .v Abstract .......................................................................... ix Chapter 1 Introduction . .1 2 Complete Metric Spaces . .8 3 The Contraction Mapping Principle . 22 4 The Barnsley Method . 40 5 Looking Forward . 48 References . 60 Vita .............................................................................. 61 iii List of Tables 1.1 Calculating Fractal Dimension for the British Coastline . .7 iv List of Figures 1.1 S0 Original Triangle . .2 1.2 First Iteration . .2 1.3 Second Iteration. .2 1.4 Third Iteration . .2 1.5 Fourth Iteration . .2 1.6 Fifth Iteration . .2 1.7 Euclidean Elements Exhibiting Self Similarity . .4 1.8 Measuring the British Coastline . .6 2.1 Hausdorff Distance Example . 11 2.2 A, Square................................................................... 13 2.3 A + for Square . 13 2.4 A, Triangle.................................................................. 14 2.5 A + for Triangle . 14 2.6 A, Bowtie................................................................... 15 2.7 A + for Bowtie . 15 3.1 S0 Original Triangle . 29 1 3.2 T (S0) after First Iteration . 29 2 3.3 T (S0) after Second Iteration . 29 3 3.4 T (S0) after Third Iteration . 29 4 3.5 T (S0) after Fourth Iteration . 30 5 3.6 T (S0) after Fifth Iteration . 30 6 3.7 T (S0) after Sixth Iteration . 30 7 3.8 T (S0) after Seventh Iteration . 30 3.9 S0 Original Square . 31 1 3.10 T (S0) after First Iteration . 31 v 2 3.11 T (S0) after Second Iteration . 32 3 3.12 T (S0) after Third Iteration . 32 4 3.13 T (S0) after Fourth Iteration . 32 5 3.14 T (S0)after Fifth Iteration . 32 6 3.15 T (S0) after Sixth Iteration . 33 7 3.16 T (S0)after Seventh Iteration . 33 3.17 S0, Original Circle. 34 1 3.18 T (S0) after First Iteration . 34 2 3.19 T (S0) after Second Iteration . 34 3 3.20 T (S0) after Third Iteration . 34 4 3.21 T (S0) after Fourth Iteration . 35 5 3.22 T (S0) after Fifth Iteration . 35 6 3.23 T (S0) after Sixth Iteration . 35 7 3.24 T (S0) after Seventh Iteration . 35 5 3.25 T (S0)...................................................................... 38 10 3.26 T (S0)..................................................................... 38 15 3.27 T (S0)..................................................................... 38 20 3.28 T (S0)..................................................................... 38 25 3.29 T (S0)..................................................................... 39 30 3.30 T (S0)..................................................................... 39 4.1 20 Iterations . 42 4.2 200 Iterations . 42 4.3 2,000 Iterations . 42 4.4 20,000 Iterations . 42 4.5 30,000 Iterations . 42 4.6 50,000 Iterations . 42 4.7 2,000 Iterations . 43 vi 4.8 50,000 Iterations . 43 4.9 Fern ........................................................................ 45 4.10 Mandelbrot Set . 46 4.11 Zoomed Mandelbrot Set . ..