fractals in nature and mathematics: from simplicity to...
TRANSCRIPT
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
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Benoıt Mandelbrot (1924-2010)
Grew up in France
Paris and CaltechEducation
IBM Fellow
Fractals
Studied “roughness” innature
Fractal Geometry
Mandelbrot Set
TED Talk link
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Clouds and Mountains
Figure: How would you measure roughness and complexity
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Trees
Figure: Self-similarity
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Lightning
Figure: Fractals - what do you see?
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Capillaries
Figure: Leaf and rivers Figure: LungsFractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 7/41
Ferns
Figure: Do you see similarity at different scales?
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Ferns - Example of Self-Similarity
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Coastlines
Figure: What is the length of the coastline?
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What are Fractals?
From fractus, - “broken”
Self-similarity
Fractional Dimension
Figure: Fractal Fern
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Fractal Trees
Figure: Fractals - self-similarity, roughness
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Fractals in Nature
Figure: Fractals -what do you see?
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Geometric Fractals - Koch Curve (1904)
Step 1 (L = 1)
Step 2
Step 3 (L = 4(13) = 4
3)Step 4
Step 5 (L = 169
)
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Geometric Fractals - Koch Curve (1904)
Step 1 (L = 1)
Step 2
Step 3 (L = 4(13) = 4
3)Step 4
Step 5 (L = 169
)
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Geometric Fractals - Koch Curve (1904)
Step 1 (L = 1)
Step 2
Step 3 (L = 4(13) = 4
3)
Step 4
Step 5 (L = 169
)
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Geometric Fractals - Koch Curve (1904)
Step 1 (L = 1)
Step 2
Step 3 (L = 4(13) = 4
3)
Step 4
Step 5 (L = 169
)
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Geometric Fractals - Koch Curve (1904)
Step 1 (L = 1)
Step 2
Step 3 (L = 4(13) = 4
3)
Step 4
Step 5 (L = 169
)
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Koch Curve - Self Similarity (L = 4n
3n →∞)
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Simple Fractals - Sierpinski Triangle
Self-similar
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Simple Fractals - Sierpinski Triangle
Self-similar
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Simple Fractals - Sierpinski Triangle
Self-similar
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Simple Fractals - Sierpinski Triangle
Self-similar
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Simple Fractals - Sierpinski Triangle
Self-similar
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Simple Fractals - Sierpinski Triangle
Self-similar
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Simple Fractals - Sierpinski Triangle
Self-similar
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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.
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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 r
=ln 4
ln 3= 1.26.
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Sierpinski Triangle - Dimension
Self-similar
D =ln 3
ln 2= 1.585
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Coastlines - Great Britain, D = 1.25
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Coastlines -North Carolina
What is the fractal dimension of the NC coast?
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Iterations
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Logistic Map - xn+1 = rxn(1− xn), given x0
Figure: r = 0.05
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Logistic Map - xn+1 = rxn(1− xn), given x0
Figure: r = 2.0
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Logistic Map - xn+1 = rxn(1− xn), given x0
Figure: r = 3.1
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Logistic Map - xn+1 = rxn(1− xn), given x0
Figure: r = 3.5
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Logistic Map - xn+1 = rxn(1− xn), given x0
Figure: r = 3.56
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Logistic Map - xn+1 = rxn(1− xn), given x0
Figure: r = 4.0
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Bifurcations of Logistic Map - xn+1 = rxn(1− xn), given x0
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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, . . . .
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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.
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Mandelbrot Set - Bulbs
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Mandelbrot Set - Bulbs
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Mandelbrot Set - Plot and Zoom
http://www.flashandmath.com/advanced/mandelbrot/
MandelbrotPlot.html
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Iterated Function Systems
Iterated Function Systems
Scalings, Rotations, andTranslations
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IFS - Turning CHAOS into a Fractal
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IFS - CHAOS
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IFS Chaos
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IFS Chaos
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Maple Leaf - Barnsley Fractals Everywhere
The Collage Theorem and Fractal Image Compression.
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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
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Fractal Landscapes - Roughness in Nature
Mountains
Clouds
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1D Midpoint Displacement Algorithm
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1D Midpoint Displacement Algorithm
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1D Midpoint Displacement Algorithm
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1D Midpoint Displacement Algorithm
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1D Midpoint Displacement Algorithm
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Diamond - Square Algorithm
Square of size 2n + 1.
Find Midpoint,adding random smallhieghts.
Create Diamond.
Edge midpoints, . . .
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Diamond - Square Algorithm
Square of size 2n + 1.
Find Midpoint,adding random smallhieghts.
Create Diamond.
Edge midpoints, . . .
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Height Maps: Clouds and Coloring
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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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