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The Mystery of Fractals!

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by

Cian Baldwin

on 5 May 2015

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Transcript of The Mystery of Fractals!

Discovered by Gaston Julia and Pierre Fatou in 1917.
They are completely self-similar.
They contain spirals and bifurcations.
They have periodicities (repeating patterns).
They can be based on the equation: z = z +
c
.
Each set is calculated for a single value of
c
.
The starting value of
z
is varied.
Complex numbers are used for the sets.
The starting value of
z
is plotted on the axes
x-axis - real part of
z
.
y-axis - imaginary part of
z
.
Values of
z
where the result does not go to infinity are in the Julia set and are plotted black.
The other values of
z
are plotted in color.
Provide examples of self-similarity.
Show infinite complexity.
Constructed by repeating a simple process.
Show branching and spiral patterns.
They are found in many things in nature:
Vegetables/Fruits
Plants
Bodies
Sea creatures
Environment
Space
They can have different patterns:
Branching - Bifurcation
Fibonacci Spiral
What are fractals?
How big and small fractals are -
You can zoom in on mathematical fractals for an infinite number of levels.
Computers can zoom in 16 orders of magnitude easily today.
The diameter of an atom is 10 meters.
The diameter of the solar system is 10 meters.
The difference between the two is 22 orders of magnitude.
Algebraic fractals -
What fractals are used for -
We can see fractals used in many areas of our everyday lives, such as:
Medicine
Science/Industry
Military
City design
Animation
Design movie landscapes
Design animal skins (dinosaurs)
Discovered by Benoit B. Mandelbrot in 1980.
It is not completely self-similar.
It contains spirals and bifurcations.
It has periodicities (repeating patterns).
It is based on the equation:
z = z + c
.
The equation is calculated for different values of
c
.
The starting value of
z
is always zero.
Complex numbers are used for the set.
c
is plotted on the axes
x-axis - real part of
c
y-axis - imaginary part of
c
THE MYSTERY
of
FRACTALS!

Mandelbrot Set
Zoomed in to 176 orders of magnitude.
This shows more complexity than the pictures zoomed in only a few orders of magnitude.
Julia Sets
Fractals are
geometric shapes
.
You have heard of many geometric shapes: squares, triangles and circles.
New area of math
They have no simple definition
Still being defined today
They are
infinitely complex
Created by
repeating a simple process
Special
self-similarity
property

Chaos theory -
Science discovered by Edward Lorenz in 1961 while studying weather.
Chaos is said to deal with the science of surprises.
Very small differences in the input to the equations can cause wildly different results.
Chaos deals with nonlinear equations like those used in the Mandelbrot set:
z = z + c
Fractal Fun Facts -
Albrecht Durer might have constructed the first fractal in 1525 in his book "Instruction in Measurement".
A spiral galaxy is the largest natural fractal.
Fractal Activity -
Construct a Sierpinski triangle with the help of the entire class.
Everyone builds their own Sierpinski triangle.
Then we combine all of the individual triangles into one large classroom-size Sierpinski triangle.
References -
Where fractals are found -
Fractals are found in
:
Nature (living/non-living things)
Algebra
Geometry
Medicine
Engineering
Fractal patterns
Branching (bifurcations)
Spirals
Fractals found in nature -
By Cian and Nolan
Geometric fractals -
Euclid of Alexandria
325-265 BC
Defined geometry in "The Elements"
(a book about geometry)



Self-Similarity
It means that no matter how far you zoom in. You always see the same shapes and patterns but smaller.
You can shrink the original shape and put it over a portion of the original shape.
Pentaflake
Spiral Galazy
Vegetables
Broccoli - Branching fractal
Romanescu - Spiral fractal
Fruits
Pineapple - Spiral fractal
Aloe - Spiral fractal
Fiddlehead Fern - Spiral fractal
Plants
Bodies
Lungs - Branching fractal
Neurons - Branching fractal
Sea Creatures
Nautilus - Spiral fractal
Sea Urchin - Spiral fractal
Enviroment
Lightning- Branching fractal
Hurricane - Spiral fractal
Rivers/Fjords -
Branching fractals
Space
Milky Way Galaxy - Spiral fractal
Sierpinski
Triangle/Carpet
Pentaflake
Koch Snowflake
Cantor Dust
Koch Curve
Sierpinski
Triangle Construction
3-D
Sierpinski Tetrahedron
(Tetrix)
3-D
Sierpinski Cube
(Menger Sponge)
Made by repeating a simple equation many times to find out if the result goes to infinity or stays finite.
They are infinitely complex.
Computers are used to help with the calculations.
Examples:
Mandelbrot Set
Julia Set
2
Mandelbrot Set
Complex Numbers
They have real and imaginary parts
An imaginary number is :



A complex number looks like:


Mandelbrot Set
Axes
Constructing the Mandelbrot Set
Julia Sets
c = -0.89 + 0.00
i
c = -0.91 + 0.31
i
c = -1.63 - 0.03
i
c = 0.03 + 0.63
i
c = 0.26 + 0.00
i
c = 0.37 + 1.08
i
c = 0.00 + 0.00
i
Medicine

Detect emphysema in lungs by using fractal patterns.
Detect cancerous tumors looking at blood vessel patterns
Science/Industry

Computer chip cooling cooling circuits.
Non-turbulent mixers used in fields such as high-precision epoxies
Military

Fractal camouflage clothing
City Design

To study the best city design to limit natural resource usage.
Rome, Italy
Suspension bridge
Cable
Cable
Construction
and
Cross-section
Cables for bridges.
Fractal antennas in cell phones and Wi-Fi systems.
Sierpinski antenna design
Cell phone with Sierpinski carpet antenna
-10
12
Mandelbrot Set
zoomed 275 orders of magnitude
Self-similarity explained
2
Chaos

Principles of chaos:
Butterfly Effect
- Input values greatly effect results
Unpredictability
- Result can't be predicted
Order/Disorder
- Study transition between order and disorder
Mixing
- Turbulance, mixed systems can't be unmixed
Feedback
- Action generates response that modifies the action
Fractals
- Infinitely complex never ending patterns
Chaos used to model systems:
In biology, economics, planetary motion
Such as weather, turbulance, fluid dynamics, traffic on roads
Chaos Explained
Chaos
Evolution of 3 starting values
Self-similarity in Geometric Fractals
Mandelbrot Set
Values of
c
that give finite
z
2
Julia and Mandelbrot Sets

Julia Sets are connected or disconnected.
When the point
z = 0
does not go to infinity (bounded) the Julia Set is connected.
When the point
z = 0
does go to infinity (unbounded) the Julia Set is disconnected.
Disconnected Julia Sets are also called Cantor Dust

Julia and Mandelbrot Sets are related for a single equation.
Values of
c
in the Mandelbrot Set give connected Julia Sets.
Full transcript