FRACTALS The Geometry of Nature

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FRACTALSThe Geometry of Nature
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By Michael Duong
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What does this look like to you?
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Georg Cantor Infinite Insanity
-Georg Cantor was born on March 3, 1845 in St.
Petersburg, Russia. He studied number theory in
the University of Berlin. He later became a
mathematics professor of the University of Halle,
a position he would remain in for the rest of his
miserable life.
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Georg Cantor Infinite Insanity
-He began to investigate the infinity. After
developing parts of set theory, Cantor wondered
what would happen if he took a line, split it in
thirds, took away the middle, and repeated. Would
this continue to infinity. YES! This was the
birth of the first mathematical monster, the
earliest form of a fractal.
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Georg Cantor Infinite Insanity
-After numerous failed attempts to solve his
famous continuum hypothesis, and the deaths of
his loved ones, Cantors trips to the insane
asylum became more and more frequent. His work
had turned him insane. -He died in 1918.
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What does this look like to you?
Usual answers -Beetle -Sting ray -Heart shaped
thingy It is actually a FRACTAL! But, what is a
FRACTAL, and who made this one?
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Benoit Mandelbrot Father of Fractals
-Benoit Mandelbrot was born on November 20, 1924
in Lithuania. His family moved to Paris, France
when he was a child, where his uncle, Szolem, a
studious mathematician, worked. -Unlike
his uncle, Benoit Mandelbrot did not inherit a
love for numbers. He was a misfit.
DUCK SITING?
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Benoit Mandelbrot Father of Fractals
-Mandelbrot was practically illiterate, and never
learned multiplication past the 5 times
tables. -One day, in geometry class, the
professor asked the students to graph an
equation. -Mandelbrot usually struggled at this
because he lacked algebra skills, but as
Mandelbrot pondered at the equation, he could
suddenly visualize the graph. He had found his
gift.
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IBM and the FRACTALS
-In the 1950s a computer company called IBM was
looking for new mathematicians and computer
programmers -Mandelbrot was hired in 1958. His
first task was to find out why there was static
in the computer telephone lines. -When graphed
for 1 day, 1 hour, and 1 second, he realized that
all the static always remained the same. He
called this self-similarity.
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Self Similarity
-characteristic in which the smaller and smaller
details of a shape have the same original
form. -Notice how as you zoom into this self
similar shape, it looks as if you always end up
in the place you originated from. Mandelbrots
static was self similar, and so are those
hands. FRACTALS are also self similar.
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Recursive Formulas
-While at IBM, Mandelbrot tried to solve a
problem presented by a young mathematician named
Gaston Julia. (recursive formula) So
substitute a number for the x on the right. Get
an answer for x on the left, and substitute that
value on the right again, etc. Using the
IBM computers, Mandelbrot graphed this
equation. He got
Right x Left x 1 2 2
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5
26
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677
677
458,330
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The Mandelbrot Set- -the graph of the equation
xx2c. Notice that as you zoom into it, it is
self similar. Mandelbrot called this kind of
shape a fractal. Before Mandelbrot,
other mathematicians called fractals monsters,
because they iterated infinitely. (Cantor,
Weierstrass, Koch, Sierpinski, Hausdorff, Julia.)
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Fractals
-a geometric figure that is created using
iteration. -a process of repeating the same
procedure over and over again. So, when we were
creating the table for the Mandelbrot Set, we
were iterating the equation xx21. We continued
the same procedure of x2 1 and substitute in the
x 5 times. So, we went through 5 iterations,
or STAGES of the fractal.
Iteration
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Examples -This is the Koch snowflake by Helge
von Koch. It starts of as a triangle, and smaller
triangles are added in the middle third of every
straight line. How many stages does the animation
go through? This is Sierpinskis Triangle by
Waclaw Sierpinski. It begins as a triangle and
continues to add black triangles inside
itself. What stage does the animation begin
at? How many stages does the animation go
through?
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Real World Fractals- Many people believe fractals
are just pretty pictures. Can you think of any
fractals in the real world? Mandelbrot
died on October 14, 2010, knowing that without
his contributions, we would still be living in
the Dark Ages.
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THE END?
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