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 BenoˆıtMandelbrot (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 → ∞)

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 complex√ 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 = {−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, 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. 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 Diamond - Square Algorithm

Square of size 2n + 1. Find Midpoint, adding random small hieghts. Create Diamond. Edge midpoints, ...

Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 38/41 Diamond - Square Algorithm

Square of size 2n + 1. Find Midpoint, adding random small hieghts. Create Diamond. Edge midpoints, ...

Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 38/41 Height Maps: Clouds and Coloring

Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 39/41 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, 2017 40/41 Conclusion

Fractals - Measure of Roughness J. Gleick, Chaos, Making a New Fractal Dimension Science, 1987/2008 Function Iteration - Mandelbrot E. Lorenz, The Essence of Set Chaos, Making a New Science, Applications 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, 2017 41/41