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Fractals

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by

Chloe Pacyna

on 14 November 2014

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Transcript of Fractals

Benoit Mandelbrot
Chloe Pacyna
Fractal Geometry
Fractals in Nature
Self Similarity
Koch Curve
Mandelbrot Set
Work Cited
Lightning: http-//www.cwu.edu/~ratlifja/Pictures/lightninggallery.jpg
Plant: http-//thereisnocavalry.files.wordpress.com/2012/08/dscn7165.jpg
Lungs: http-//classes.yale.edu/fractals/panorama/Biology/Physiology/Lung1.gif
Mandelbrot: http-//users.math.yale.edu/mandelbrot/BBM-2000-hires.jpg
Mandelbrot Set Explanation: http-//www.butterflyeffect.ca/Close/Images/MSetHow.jpg
Fractal Beginnings:
Cantor Set
Cantor Set: http://jwilson.coe.uga.edu/EMAT6680Fa11/Frailey/FractalEssay.htm
How Fractals Work
y=2x+1
x
y
1
3
3
7
7
15
15
31
Fractal Zoom
Heart Rate Variability
Heart: http://heartresearch.org.uk/sites/default/files/pageimages/Heart%20iStock%20pic%20(1).jpg
Urban Sprawl
City 2: http://3.bp.blogspot.com/-yZk1OZ-FbB8/T1c61-SqRyI/AAAAAAAACTo/uoLqj0aHJBo/s1600/urban+fractals+93.bmp
City 1: http://fc03.deviantart.net/fs22/f/2007/314/8/8/City_Map_2_by_Ometheon.jpg
Fractal Cancer Diagnostics
Cancer: http://www.junkdna.com/nova_fractal_cancer_2008.JPG
HRV: http://zoepercavia.com/media/images/figure1_76C82B187A61E6D8AB7605D95525449E.png
Main Mandelbrot: http://upload.wikimedia.org/wikipedia/commons/2/21/Mandel_zoom_00_mandelbrot_set.jpg
Sierpinski: https://www.zeuscat.com/andrew/chaos/sierpinski.html
Fractals
Innovative
Beautiful
Helpful
Mandelbrot Set
fc(z)=z^2+c
z
fc(z)
when c=1, looking at behavior of 0
0
0^2+1=1
1
1^2+1=2
2^2+1=5
2
5
5^2+1=26
goes to infinity!
What about bounded numbers (never larger than 2)?
fc(z)=z^2+1
where c=-1, looking at behavior of 0
z
fc(z)
0
0^2-1=-1
-1
(-1)^2-1=0
0
0^2-1=-1
-1
(-1)^2-1=0
Never strays from [-1,0]!
We call anything that stays within 2 units of itself bounded
Mandelbrot Set
Any number that remains close to where it starts (within 2) is in the Mandelbrot set (colored black)
Any numbers that zoom to infinity will be sorted into groups of going to infinity at different rates (colored different shades)
Financial Fractals
Fractals in Forestry and CO2 Studies
Market: http://www.tradersnarrative.com/wp-content/uploads/2009/06/sensex%20chart%20ewi%20market%20fractal.jpg
Forest 1: http://freebigpictures.com/
Forest 2: http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAcQjRw&url=http%3A%2F%2Fwww.ca.uky.edu%2Fforestry%2Fmaehrbearky.php&ei=cYZlVKjlLYiYyASy6oDAAQ&bvm=bv.79142246,d.aWw&psig=AFQjCNEx7KIAK6aBFP2rEqOxV33gVW28Ag&ust=1416025943100412
Full transcript