Fractals in Nature and Mathematics: From Simplicity to Complexity Dr. R. L. Herman, UNCW Mathematics & Physics Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 1/41 Outline 1 Complexity in Nature 2 What are Fractals? 3 Geometric Fractals 4 Fractal Dimensions 5 Function Iteration 6 Applications Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 2/41 Beno^ıtMandelbrot (1924-2010) Grew up in France Paris and Caltech Education IBM Fellow Fractals Studied \roughness" in nature Fractal Geometry Mandelbrot Set TED Talk link Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 3/41 Clouds and Mountains Figure: How would you measure roughness and complexity Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 4/41 Trees Figure: Self-similarity Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 5/41 Lightning Figure: Fractals - what do you see? Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 6/41 Capillaries Figure: Leaf and rivers Figure: Lungs Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 7/41 Ferns Figure: Do you see similarity at different scales? Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 8/41 Ferns - Example of Self-Similarity Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 9/41 Coastlines Figure: What is the length of the coastline? Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 10/41 What are Fractals? From fractus, - \broken" Self-similarity Fractional Dimension Figure: Fractal Fern Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 11/41 Fractal Trees Figure: Fractals - self-similarity, roughness Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 12/41 Fractals in Nature Figure: Fractals -what do you see? Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 13/41 Step 2 1 4 Step 43 (L = 4( 3 ) = 3 ) 16 Step 5 (L = 9 ) Geometric Fractals - Koch Curve (1904) Step 1 (L = 1) Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 14/41 1 4 Step 43 (L = 4( 3 ) = 3 ) 16 Step 5 (L = 9 ) Geometric Fractals - Koch Curve (1904) Step 1 (L = 1) Step 2 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 14/41 Step 4 16 Step 5 (L = 9 ) Geometric Fractals - Koch Curve (1904) Step 1 (L = 1) Step 2 1 4 Step 3 (L = 4( 3 ) = 3 ) Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 14/41 1 4 Step 3 (L = 4( 3 ) = 3 ) 16 Step 5 (L = 9 ) Geometric Fractals - Koch Curve (1904) Step 1 (L = 1) Step 2 Step 4 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 14/41 1 4 Step 3 (L = 4( 3 ) = 3 ) Geometric Fractals - Koch Curve (1904) Step 1 (L = 1) Step 2 Step 4 16 Step 5 (L = 9 ) Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 14/41 4n Koch Curve - Self Similarity (L = 3n ! 1) Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 15/41 Self-similar Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 16/41 Self-similar Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 16/41 Self-similar Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 16/41 Self-similar Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 16/41 Self-similar Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 16/41 Self-similar Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 16/41 Simple Fractals - Sierpinski Triangle Self-similar Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 16/41 Dimensions - r = magnification, n = Number of shapes n = 2; r = 2 ) ln 2 = ln 2: n = 4; r = 2 ) ln 4 = 2 ln 2: n = 8; r = 2 ) ln 8 = 3 ln 2: ln n D = : ln r Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 17/41 Fractal Dimensions - Koch Curve For each step length of line segment is reduced by r = 3: The number of lines increases by factor n = 4: Therefore ln n D = ln r ln 4 = ln 3 = 1:26: Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 18/41 Sierpinski Triangle - Dimension Self-similar ln 3 D = = 1:585 ln 2 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 19/41 Coastlines - Great Britain, D = 1:25 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 20/41 Coastlines -North Carolina What is the fractal dimension of the NC coast? Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 21/41 Iterations Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 22/41 Logistic Map - xn+1 = rxn(1 − xn); given x0 Figure: r = 0:05 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 23/41 Logistic Map - xn+1 = rxn(1 − xn); given x0 Figure: r = 2:0 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 23/41 Logistic Map - xn+1 = rxn(1 − xn); given x0 Figure: r = 3:1 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 23/41 Logistic Map - xn+1 = rxn(1 − xn); given x0 Figure: r = 3:5 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 23/41 Logistic Map - xn+1 = rxn(1 − xn); given x0 Figure: r = 3:56 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 23/41 Logistic Map - xn+1 = rxn(1 − xn); given x0 Figure: r = 4:0 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 23/41 Bifurcations of Logistic Map - xn+1 = rxn(1 − xn); given x0 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 24/41 Mandelbrot Set Iterate complexp numbers z = a + bi; i = −1: 2 zn+1 = zn + c; z0 = 0: Example: c = 1 : 0; 1; 2; 5; 26 :::: Example: c = −1 : 0; −1; 0; −1; 0;:::: Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 25/41 2 Mandelbrot Set - zn+1 = zn + c; z0 = 0: Example: c = i : x0 = 0 2 x1 = 0 + i = i 2 x2 = i + i = −1 + i 2 x3 = (−1 + i) + i = −i 2 x4 = (−i) + i = −1 + i 2 x4 = (−i) + i = −i Gives period 2 orbit = f−1 + i; −i; −1 + i; −i;::: g. Show more points at Fractals site. Mandelbrot Set Zoom online. Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 26/41 Mandelbrot Set - Bulbs Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 27/41 Mandelbrot Set - Bulbs Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 28/41 Mandelbrot Set - Plot and Zoom http://www.flashandmath.com/advanced/mandelbrot/ MandelbrotPlot.html Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 29/41 Iterated Function Systems Iterated Function Systems Scalings, Rotations, and Translations Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 30/41 IFS - Turning CHAOS into a Fractal Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 31/41 IFS - CHAOS Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 32/41 IFS Chaos Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 33/41 IFS Chaos Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 33/41 Maple Leaf - Barnsley Fractals Everywhere The Collage Theorem and Fractal Image Compression. Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 34/41 The Genesis Effect - Star Trek II: The Wrath of Khan (1982) Fractal Landscapes First completely computer-generated sequence in a film http: //design.osu.edu/carlson/ history/tree/images/ pages/genesis1_jpeg.htm https://www.youtube.com/ watch?v=QXbWCrzWJo4 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 35/41 Fractal Landscapes - Roughness in Nature Mountains Clouds Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 36/41 1D Midpoint Displacement Algorithm Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 37/41 1D Midpoint Displacement Algorithm Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 37/41 1D Midpoint Displacement Algorithm Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 37/41 1D Midpoint Displacement Algorithm Fractals in Nature and Mathematics R.