Fractals in Nature and Mathematics: From Simplicity to Complexity
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1 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, /41
2 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, /41
3 Benoît Mandelbrot ( ) 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, /41
4 Clouds and Mountains Figure: How would you measure roughness and complexity Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
5 Trees Figure: Self-similarity Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
6 Lightning Figure: Fractals - what do you see? Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
7 Capillaries Figure: Leaf and rivers Figure: Lungs Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
8 Ferns Figure: Do you see similarity at different scales? Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
9 Ferns - Example of Self-Similarity Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
10 Coastlines Figure: What is the length of the coastline? Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
11 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, /41
12 Fractal Trees Figure: Fractals - self-similarity, roughness Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
13 Fractals in Nature Figure: Fractals -what do you see? Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
14 Geometric Fractals - Koch Curve (1904) Step 1 (L = 1) Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
15 Geometric Fractals - Koch Curve (1904) Step 1 (L = 1) Step 2 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
16 Geometric Fractals - Koch Curve (1904) Step 1 (L = 1) Step 2 Step 3 (L = 4( 1 3 ) = 4 3 ) Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
17 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, /41
18 Geometric Fractals - Koch Curve (1904) Step 1 (L = 1) Step 2 Step 4 Step 5 (L = 16 9 ) Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
19 Koch Curve - Self Similarity (L = 4n 3 n ) Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
20 Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
21 Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
22 Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
23 Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
24 Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
25 Simple Fractals - Sierpinski Triangle Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
26 Simple Fractals - Sierpinski Triangle Self-similar Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
27 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. D = ln n ln r. Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
28 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 D = ln n ln r = ln 4 ln 3 = Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
29 Sierpinski Triangle - Dimension Self-similar D = ln 3 ln 2 = Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
30 Coastlines - Great Britain, D = 1.25 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
31 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, /41
32 Iterations Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
33 Logistic Map - x n+1 = rx n (1 x n ), given x 0 Figure: r = 0.05 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
34 Logistic Map - x n+1 = rx n (1 x n ), given x 0 Figure: r = 2.0 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
35 Logistic Map - x n+1 = rx n (1 x n ), given x 0 Figure: r = 3.1 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
36 Logistic Map - x n+1 = rx n (1 x n ), given x 0 Figure: r = 3.5 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
37 Logistic Map - x n+1 = rx n (1 x n ), given x 0 Figure: r = 3.56 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
38 Logistic Map - x n+1 = rx n (1 x n ), given x 0 Figure: r = 4.0 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
39 Bifurcations of Logistic Map - x n+1 = rx n (1 x n ), given x 0 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
40 Mandelbrot Set Iterate complex numbers z = a + bi, i = 1. z n+1 = z 2 n + c, z 0 = 0. Example: c = 1 : 0, 1, 2, 5, Example: c = 1 : 0, 1, 0, 1, 0,.... Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
41 Mandelbrot Set - z n+1 = z 2 n + c, z 0 = 0. Example: c = i : x 0 = 0 x 1 = i = i x 2 = i 2 + i = 1 + i x 3 = ( 1 + i) 2 + i = i x 4 = ( i) 2 + i = 1 + i x 4 = ( i) 2 + i = i Gives period 2 orbit = { 1 + i, i, 1 + i, i,... }. Show more points at Fractals site. Mandelbrot Set Zoom online. Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
42 Mandelbrot Set - Bulbs Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
43 Mandelbrot Set - Bulbs Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
44 Mandelbrot Set - Plot and Zoom MandelbrotPlot.html Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
45 Iterated Function Systems Iterated Function Systems Scalings, Rotations, and Translations Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
46 IFS - Turning CHAOS into a Fractal Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
47 IFS - CHAOS Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
48 IFS Chaos Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
49 IFS Chaos Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
50 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, /41
51 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 watch?v=qxbwcrzwjo4 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
52 Fractal Landscapes - Roughness in Nature Mountains Clouds Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
53 1D Midpoint Displacement Algorithm Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
54 1D Midpoint Displacement Algorithm Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
55 1D Midpoint Displacement Algorithm Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
56 1D Midpoint Displacement Algorithm Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
57 1D Midpoint Displacement Algorithm Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
58 Diamond - Square Algorithm Square of size 2 n + 1. Find Midpoint, adding random small hieghts. Create Diamond. Edge midpoints,... Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
59 Diamond - Square Algorithm Square of size 2 n + 1. Find Midpoint, adding random small hieghts. Create Diamond. Edge midpoints,... Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
60 Height Maps: Clouds and Coloring Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
61 Other Applications Video Games Fracture - link Ceramic Material - link Biology - wrinkles, lungs, brain,... Astrophysics Figure: The Sloan Digital Sky Survey Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
62 Conclusion Fractals - Measure of Roughness Fractal Dimension Function Iteration - Mandelbrot Set Applications J. Gleick, Chaos, Making a New Science, 1987/2008 E. Lorenz, The Essence of Chaos, Making a New Science, 1995 B. Mandelbrot, The Fractal Geometry of Nature, 1982 Barnsley, Fractals Everywhere, 1988/2012 Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, /41
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