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The Mandelbrot Set: A Fractal Masterpiece
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TweetMatty Cochran
on 6 May 2014Transcript of The Mandelbrot Set: A Fractal Masterpiece
The Mandelbrot Set: A Fractal Masterpiece
By:Dylan Acevedo and Matt Weinsting
Benoit B. Mandelbrot discovered the Mandelbrot set in 1980.
Came from the interest in Mathematical monsters.
The Mandelbrot Set
Needed Basics
Iteration
Complex numbers >
The complex plane
Multiplying complex numbers
Complex Plane
Making the Julia Set pictures
The pictures are made by coloring complex coordinates depending on how they tend to infinity, otherwise known as the coordinates
fate.
Coloring
How fast the coordinate tends to infinity determines what color it is assigned.
Colors associated with coordinates that tend to infinity range from red, orange, yellow, green, blue, and violet.
Red being very quickly and violet being slow.
If it tends to never go to infinity it is given black.
Julia Sets
Interesting facts
Mandelbrot set and fractal geometry was discredited by mathematicians as useless math
His work would turn out, though, to be very applicable in the real world
Real world applications
Biology (bronchi, tumor blood flow)
Tree formations
Computer generated animations
Antennas(fractal patterns led to smaller devices)
These are just some of its influences
The purpose of this presentation
To understand how Julia sets are made, specifically how the pictures are made and how they are related to the formation of the Mandelbrot set. As well as examine a proof that describes the boundaries of Julia Sets and the Mandelbrot Set
Julia sets cont...
Julia sets are defined as either being disconnected or connected.
Julia set categories
How to determine a Julia Set's category
A mathematical proof has been made to determine what category a Julia set is by checking the fate of 0 using the specific Julia set equation.
Consider a Julia Set
Relating the two categories
Consider a new complex plane.
start by plotting 1+.5i
Notice that when iterating 0, it tended to infinity relatively quickly.
So we color the coordinate red.
Cont...
Now consider every other c value and its fate of 0.
Once you color (almost) every other c value depending on its fate of 0 you reach...
The infamous Mandelbrot Set!
Determining fate
Coordinates are picked from the complex plane and iterated using the equation:
Consider this complex coordinate:
Disconnected
Connected
Why do we care?
Julia Sets in the Mandelbrot Set
http://www.oxfordstrat.com/resources/ideas/mandelbrotbenoit/
http://lbc9.com/digitalart/fractals/mandelbrotset/wallpapers11.html
 i

1
1


2
 i
Where z is a complex variable, and c is a fixed complex number.
http://www.math.ucr.edu/home/baez/roots/
For Further Research
http://math.bu.edu/individual/bob/index.html
Questions?
Modulus
Modulus is the length of the coordinate from the origin
If the modulus of
z
or
c
is greater than 2, then the iteration process will always go to infinity.
Theorems needed to prove the modulus proof
Theorem 2.17
Proof of Theorem 2.17
Proof cont.
Corollary 2.18
The proof of this corollary can be looked at for further study.
Let
(z(n))
be the sequence of iterates of the function for If the modulus of
z
, 
z
, is greater than 2 and 
c
 less than or equal to 2, then...
Full transcriptBy:Dylan Acevedo and Matt Weinsting
Benoit B. Mandelbrot discovered the Mandelbrot set in 1980.
Came from the interest in Mathematical monsters.
The Mandelbrot Set
Needed Basics
Iteration
Complex numbers >
The complex plane
Multiplying complex numbers
Complex Plane
Making the Julia Set pictures
The pictures are made by coloring complex coordinates depending on how they tend to infinity, otherwise known as the coordinates
fate.
Coloring
How fast the coordinate tends to infinity determines what color it is assigned.
Colors associated with coordinates that tend to infinity range from red, orange, yellow, green, blue, and violet.
Red being very quickly and violet being slow.
If it tends to never go to infinity it is given black.
Julia Sets
Interesting facts
Mandelbrot set and fractal geometry was discredited by mathematicians as useless math
His work would turn out, though, to be very applicable in the real world
Real world applications
Biology (bronchi, tumor blood flow)
Tree formations
Computer generated animations
Antennas(fractal patterns led to smaller devices)
These are just some of its influences
The purpose of this presentation
To understand how Julia sets are made, specifically how the pictures are made and how they are related to the formation of the Mandelbrot set. As well as examine a proof that describes the boundaries of Julia Sets and the Mandelbrot Set
Julia sets cont...
Julia sets are defined as either being disconnected or connected.
Julia set categories
How to determine a Julia Set's category
A mathematical proof has been made to determine what category a Julia set is by checking the fate of 0 using the specific Julia set equation.
Consider a Julia Set
Relating the two categories
Consider a new complex plane.
start by plotting 1+.5i
Notice that when iterating 0, it tended to infinity relatively quickly.
So we color the coordinate red.
Cont...
Now consider every other c value and its fate of 0.
Once you color (almost) every other c value depending on its fate of 0 you reach...
The infamous Mandelbrot Set!
Determining fate
Coordinates are picked from the complex plane and iterated using the equation:
Consider this complex coordinate:
Disconnected
Connected
Why do we care?
Julia Sets in the Mandelbrot Set
http://www.oxfordstrat.com/resources/ideas/mandelbrotbenoit/
http://lbc9.com/digitalart/fractals/mandelbrotset/wallpapers11.html
 i

1
1


2
 i
Where z is a complex variable, and c is a fixed complex number.
http://www.math.ucr.edu/home/baez/roots/
For Further Research
http://math.bu.edu/individual/bob/index.html
Questions?
Modulus
Modulus is the length of the coordinate from the origin
If the modulus of
z
or
c
is greater than 2, then the iteration process will always go to infinity.
Theorems needed to prove the modulus proof
Theorem 2.17
Proof of Theorem 2.17
Proof cont.
Corollary 2.18
The proof of this corollary can be looked at for further study.
Let
(z(n))
be the sequence of iterates of the function for If the modulus of
z
, 
z
, is greater than 2 and 
c
 less than or equal to 2, then...