Fractals in Nature and Mathematics: From Simplicity to...

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 Nature2 What are Fractals?3 Geometric Fractals4 Fractal Dimensions5 Function Iteration6 Applications

Benoıt Mandelbrot (1924-2010)

Grew up in France

Paris and CaltechEducation

IBM Fellow

Fractals

Studied “roughness” innature

Fractal Geometry

Mandelbrot Set

TED Talk link

Clouds and Mountains

Figure: How would you measure roughness and complexity

Figure: Self-similarity

Lightning

Figure: Fractals - what do you see?

Capillaries

Figure: Leaf and rivers Figure: LungsFractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 7/41

Figure: Do you see similarity at different scales?

Ferns - Example of Self-Similarity

Coastlines

Figure: What is the length of the coastline?

What are Fractals?

From fractus, - “broken”

Self-similarity

Fractional Dimension

Figure: Fractal Fern

Fractal Trees

Figure: Fractals - self-similarity, roughness

Fractals in Nature

Figure: Fractals -what do you see?

Geometric Fractals - Koch Curve (1904)

Step 1 (L = 1)

Step 2

Step 3 (L = 4(13) = 4

3)Step 4

Step 5 (L = 169

Step 1 (L = 1)

Step 2

Step 3 (L = 4(13) = 4

3)Step 4

Step 5 (L = 169

Step 1 (L = 1)

Step 2

Step 3 (L = 4(13) = 4

Step 4

Step 5 (L = 169

Step 1 (L = 1)

Step 2

Step 3 (L = 4(13) = 4

Step 4

Step 5 (L = 169

Step 1 (L = 1)

Step 2

Step 3 (L = 4(13) = 4

Step 4

Step 5 (L = 169

Koch Curve - Self Similarity (L = 4n

3n →∞)

Simple Fractals - Sierpinski Triangle

Self-similar

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

Fractal Dimensions - Koch Curve

For each step length of linesegment is reduced by r = 3.

The number of lines increases byfactor n = 4.

Therefore

D =ln n

ln 3= 1.26.

Sierpinski Triangle - Dimension

Self-similar

D =ln 3

ln 2= 1.585

Coastlines - Great Britain, D = 1.25

Coastlines -North Carolina

What is the fractal dimension of the NC coast?

Iterations

Logistic Map - xn+1 = rxn(1− xn), given x0

Figure: r = 0.05

Figure: r = 2.0

Figure: r = 3.1

Figure: r = 3.5

Figure: r = 3.56

Figure: r = 4.0

Bifurcations of Logistic Map - xn+1 = rxn(1− xn), given x0

Mandelbrot Set

Iterate complex numbersz = a + bi , i =

√−1.

zn+1 = z2n + c , z0 = 0.

Example: c = 1 :

0, 1, 2, 5, 26 . . . .

Example: c = −1 :

0,−1, 0,−1, 0, . . . .

Mandelbrot Set - zn+1 = z2n + c , z0 = 0.

Example: c = i :

x0 = 0

x1 = 02 + i = i

x2 = i2 + i = −1 + i

x3 = (−1 + i)2 + i = −ix4 = (−i)2 + i = −1 + i

x4 = (−i)2 + i = −i

Gives period 2 orbit ={−1 + i ,−i ,−1 + i ,−i , . . . }.

Show more points at Fractals site.

Mandelbrot Set Zoom online.

Mandelbrot Set - Bulbs

Mandelbrot Set - Plot and Zoom

http://www.flashandmath.com/advanced/mandelbrot/

MandelbrotPlot.html

Iterated Function Systems

Scalings, Rotations, andTranslations

IFS - Turning CHAOS into a Fractal

IFS - CHAOS

IFS Chaos

Maple Leaf - Barnsley Fractals Everywhere

The Collage Theorem and Fractal Image Compression.

The Genesis Effect - Star Trek II: The Wrath of Khan(1982)

Fractal Landscapes

First completelycomputer-generated sequence ina film http:

//design.osu.edu/carlson/

history/tree/images/

pages/genesis1_jpeg.htm

https://www.youtube.com/

watch?v=QXbWCrzWJo4

Fractal Landscapes - Roughness in Nature

Mountains

Clouds

1D Midpoint Displacement Algorithm

Diamond - Square Algorithm

Square of size 2n + 1.

Find Midpoint,adding random smallhieghts.

Create Diamond.

Edge midpoints, . . .

Diamond - Square Algorithm

Square of size 2n + 1.

Find Midpoint,adding random smallhieghts.

Create Diamond.

Edge midpoints, . . .

Height Maps: Clouds and Coloring

Other Applications

Video Games

Fracture - link

Ceramic Material - link

Biology - wrinkles, lungs,brain, ...

Astrophysics

Figure: The Sloan Digital Sky SurveyFractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 40/41

Conclusion

Fractals - Measure of Roughness

Fractal Dimension

Function Iteration - MandelbrotSet

Applications

J. Gleick, Chaos, Making a NewScience, 1987/2008

E. Lorenz, The Essence ofChaos, Making a New Science,1995

B. Mandelbrot, The FractalGeometry of Nature, 1982

Barnsley, Fractals Everywhere,1988/2012

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