Transcript Diapositiva 1

The concept…
Fractal Geometry , also called the “true geometry of
nature” could be explained as: non-triangular or squared shaped
figures, but as more complex ones. Fractals are used to explain
concrete objects.
For example : in regular objects as in nature, we need fractal
geometry. Fractal geometry deals with non-integer dimensions.
Zero dimensional point is only a point. One is a Line or a
curve , two dimensional figures could be plane figures as circles ,
triangles or squares. Three dimensional solid figures are solids such
as cubes, pyramids, cylinders and spheres. (three dimensional
figures). Which are used just for a model of reality.
Unlike the other dimensions, the fourth one is
located in the real world, in the one we live.
Continuation
We are actually living in the fourth dimension of the “spacetime continuum “
“Fractals are an open system where everything is related to
everything else. “
Before fractal geometry there was Euclidian geometry which
was used thousands of years ago but it only stated artificial
realities, such as: the fist , second and third dimension. These
dimensions are imaginary, only the fourth is real.
These caused great advance in math. Math now, is a
complete subject , separated completely by physics (science)
its sister subject for centuries.
Fractals were discovered in the decade of the 1970’s.
What are images?
Images are geometric figures in
irregular shape in form of pattern that
repeats itself with similarity; sometimes
even with exactitude in an endless
manner having a bigger image, being
repeated into smaller ones, and smaller
and each time smaller, until they are
practically invisible to the human eye.
Example of images:
Julia set with
cousines
Mandelbrot set
Julia set
A Mandelbrot shape
wich
apears
in
seahorse valley after
magnification
Sierpinski’s
triangle
Some images
Some images
Uses of fractal Geometry: Medicine
(Science)
Fractals are found in the body. Fractals can help us to
understand better the human body. The best known
example are the arteries and veins in mammalian vascular
systems. “This area of biological research is just beginning.
Chaotician Michael McGuire refers to recent discoveries in
brain research which suggests that a fractal structure based
on hexagons may be how the receptive fields of the visual
cortex are organized. The smallest hexagons correspond to
the cells of the retina and perception of fine details,
the larger hexagons organize the underlying layers to
recognize progressively coarse detail.”
Entratainment industry
Fractal geometry is used also in
entertainment, (technology) industry . Fractal
geometry can be used today in movie making ,
principally in the especial effects and animation
in three dimensional cartoons, like for example:
Jimmy Neutron and The ice age, which where
only created with fractals. Thanks to these, we are
leading to a new generation of cartoons and
television series.
Music industry
Fractal Geometry can also be used in music
industry. In order to a song , to have a certain rhythm,
it has to repeat its self over and over, with out losing
the original beat, therefore, fractal geometry is used
for music. Also for the “music waves” and for the
images, fractal geometry is used. Midi files, which are
the ones for the use on internet web pages, are
especially formed with fractals and Iteration because
they never end, they are design to keep up on going
and going, until the user closes the window.
Philosophy and …nature
Philosophy are mainly referring at
the study of men, how they think and to
think about explanations for things. Well
guess, what know fractal geometry is also
affecting this because fractal geometry is
the why about everything in nature, the
way it is. It’s mainly telling everything with
math. Why flower are like that or why
snowflakes are awesome, because there’s
never another, exactly the same , they have
infinite patterns.
Science: space, meteorology
Space is an excellent example to describe
Fractals. What we’ve noticed about space we can
state that are fractals. An image of space can be
revealed as a fractal image. We, human don’t
even know were the universe ends. Therefore,
we can state it as a fractal, fractals many times
don’t end, they just repeat, just as space.
Meteorologists
apply
fractals,
to
comprehend the form, shape and even the
composition of natural bodies such are clouds
or as already said, snowflakes.
Biography: Benoit Mandelbrot
BenoitMandelbrot.
He
is
responsible for the actual interest in
fractal geometry. He showed how
fractals can happen in many different
places in both: mathematics and
elsewhere in nature.
Mandelbrot was born in Poland in 1924.
His family had a very academic tradition. His
father, however, made his living buying and
selling clothes while his mother was a doctor. As
a young boy, Mandelbrot was introduced to
mathematics by his two uncles.
Continuation…
Mandelbrot's family emigrated to France in
1936. His uncle Szolem Mandelbrojt, who was
Professor of Mathematics at the College de France
and successor of Hadamard in this post, took
responsibility for his education.
Mandelbrot attended “Lycée Rolin” in Paris up till the
start of the second World War. This was a very difficult
period during
Mandelbrot’s life, who feared for his life
on various occasions.
Thanks to his unconventional education ,
Mandelbrot thought in other ways that would be
difficult for someone that had an conventional
education.
Then Mandelbrot went to study at Lyon , Paris.
There, Paul Lévy teached Mandelbrot and became one
strong influence for him. Then he went to Princeton
for the advance study of technology, and he was
sponsored by John von Neumann.
Mandelbrot went back to France in 1955 and
worked at the Centre National de la Recherche
Scientific. There, he married Aliette Kagan, he didn’t
stay there too long before returning to the United
States. He was unhappy. In 1945 Mandelbrot's uncle
had introduced him to Julia’s important 1918 paper
claiming.
Mandelbrot was asked to work for the IBM
company (in the watson reserch center), there he had a
wonderful space of work, and he like it , besides there he
could do his reserch.
Mandelbrot met Julia, and later he published
Julia's work . Jualia’s work is now known as the most
beautiful fractals of today. His work was printed in his
book “Les objets fractals, forn, hasard et dimension”
(1975) and more complete in The fractal geometry of
nature in 1982. On 23 June 1999 Mandelbrot got the
Honorary Degree of Doctor of Science from the
University of St Andrews.
While he was an IBM fellow, he was a professor
of mathematics at Harvard University and ´Ecole
Polytechnique , professor of engineering at Yale,
Professor of Economics at Harvard, and of Professor of
Physiology at the Einstein College of Medicine.
“Mandelbrot has received numerous honors and
prizes in recognition of his remarkable achievements. Just
for example, in 1985 Mandelbrot was awarded the 'Barnard
Medal for Meritorious Service to Science'. The following
year he received the Franklin Medal. In 1987 he was
honored with the Alexander von Humboldt Prize, receiving
the Steinmetz Medal in 1988 and many more awards
including the Nevada Medal in 1991 and the Wolf prize for
physics in 1993”.
Mandelbrot also discovered Madelbrot set.
Which is a simple formula: z -> z^2 + c , even with
computers, the formula couldn't had been discovered
without him. With these formula , the world has been
done a great change. Benoit Mandelbrot, is now an
IBM scientist and Professor of Mathematics at Yale.
Iteration, what is it? How does it
relate?
Many sets, principally the Mandelbrot set
are generated by iteration. Iteration means to
repeat a process over and over again. In
mathematics, this process is most often the use of a
mathematical function. For example: “for the
Mandelbrot set, the function involved is the
simplest nonlinear function imaginable, namely x2
+ c, where c is a constant. ”The answer has been got
from an specific and repetitive order of operations.
This formula at the end, it tends to Infinity,
therefore, it has to be a lot with fractals.
Like as I mentioned at the beginning: “Fractals are
endless repetitive patterns .” so, Iteration makes up
the repetitive part of fractals, besides , Iteration
makes fractals more precise.
How are imaginary numbers related?
Although “imaginary numbers” are not imaginary; in
fact they a very real, these numbes are very useful for the fractal
geometry.
In fractal geometry people coulnd’t get the
fractals right because real numbers are just “big”. With
imaginary numbers people can’t state how many litters
are in a bottle of water or how meny kilos are in a bag.
Imaginary numbers are used to calculate things that
can´t be calculated with “real” numbers. Imaginary
numbers are related to fractals, in the way that fractals
can’t be calculate with real numbers, therefore we need
the “imaginary numbers” with out them , fractal
geometry wouldn’t be possible.
Conclusions:
Fractals are extremely important. Fractals are just another way to
see life. Everything has to do with them , even the most little thing in the
world that I can think of. Fractals are the future of humans, fractals are
leading humans to a complete new generation. There is no doubt that
thanks to Mandelbrot's discovery , the human specie is going to advance
way too much.
I liked working on this project more than other I had worked on
in math. I think this project is very original, and different from the
others. This has a clear point and I really learned about it because before I
didn’t had a clue about what fractal geometry or a fractal was. I practically
learned all I wrote. It really caught my attention. This helped me to
understand imaginary numbers better, because now I can say
that imaginary numbers are actually non-imaginary numbers
that complement a very important part of our world.
Bibliography:
•http://home.inreach.com/kfarrell/fractals.html
•http://math.rice.edu/~lanius/fractals/WHY/
•http://srd.yahoo.com/S=2766679/K=fractal+geometry+%2b+its+uses/v=2/l=WS1/R
=3/H=0/*-http://www.fractalwisdom.com/FractalWisdom/fractal.html
•http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html
•http://www.io.com/~mgscheue/chaos/iteration.html
•http://hilbert.dartmouth.edu/~doyle/docs/icos/icos/icos.html
•http://hilbert.dartmouth.edu/~doyle/docs/icos/icos/icos.html
•http://astronomy.swin.edu.au/~pbourke/fractals/
•http://srd.yahoo.com/S=2766679/K=imaginary+numbers/v=2/l=WS1/R=1/SS=8865752
/H=0/*-http://www.math.toronto.edu/mathnet/answers/imaginary.html