The following presentation describes what Fractals are, the history of Fractals, and
how fractals relate to global warming, medicine, nature, and art. A fractal is a curve or geometric
figure, each part of which has the same statistical character as the whole. Fractals are useful
when looking at patterns that recur in progressively smaller scales and in describing partly random
or chaotic phenomena. A simple fractal often observed when trying to understand this concept
would be the Koch snowflake. The Koch snowflake is so named for its Mathematician maker, Helge
von Koch, who was born January 25, 1870 in Stockholm Sweden. Illustrations of the iterative,
progressions of the snowflake are demonstrated here.This is a demonstration of how to create
the Koch Snowflake. This fractal starts as an equilateral/equiangular triangle. An equilateral/equiangular
triangle is a three sided figure in which all sides are of equal length and all angles
have a measure of sixty degrees. From this point the progression begins. Remove the inner
third of each side of the triangle and build another equilateral triangle where the middle
thirds were removed. This makes the second progression. The third progression continues
in a similar fashion. Replace the inner third of each side with another equilateral triangle.
This process can be repeated an infinite number of times. A key feature that applies to all
fractals, seen in the snowflake example, is self-similarity. In other words, if you zoom
in on a very small portion of the fractal, it looks very similar to every other part
of the fractal. A tree can be described in this way. When zooming into a part of a tree,
such as a branch with twigs, it would look very similar to the trunk and its branches.
The illustrations demonstrate the self-similarity a tree might have. Each progression from left
to right zooms more closely than the previous image, yet each image still looks very similar
to the last. This self-similarity can be seen in many aspects of nature from rivers, to
mountains, to seashells and to clouds. Another fractal that is well known is the Cantor Set
or Dust. The creator, Georg Cantor, was born March 3, 1845 in St. Petersburg, Russia. He
became fascinated with the fact that there are an infinite number of points on any particular
line segment. The Cantor Set attempts to demonstrate what happens when an infinite number of line
segments are removed from an initial line interval. The Cantor Set starts with the interval
[0,1], which means, the points 0 and 1 are included in the interval, as well as every
point between. Then, from this line segment, remove the middle third or the interval [1/3,
2/3]. This dust is now called a triadic set. Then, with the two new lines present, remove
the middle third from each. The Sierpinski Gasket, was created by the famous Mathematician,
Waclaw Sierpinski, who was born March 14, 1882 in Warsaw Poland. We will explore 3 ways
of creating the Sierpinski Gasket. One way to make this fractal, is by starting with
a flat plane consisting of points P1, P2, and P3. Then, pick a point anywhere on the
plane. Now randomly choose any of the original 3 points and draw an infinitely small point
in the middle of the distance between the chosen point and the original point. Continue
this process using the new infinitely small point and another randomly chosen original
point. All the points in this generation lay within
the triangular boundary that points P1, P2, and P3 make. However, the strange part is
that there are large sections within the triangle that never contain any points. The distribution
of points form a 2D Sierpinski Gasket as shown, each image from left to right containing more
and more points. Another way to generate a 2D version of the Sierpinski Gasket is by
starting with a triangle and cutting a new triangle out of the middle, resulting in the
creation of 3 solid triangles and one cut out triangle. The triangle that is cut-out
was formed by selecting the midpoints on each side of the original triangles. Then, those
midpoints were used as the vertices of the inscribed triangle. Repeat this process with
the 3 solid remaining triangles. The third method of creating the Sierpinski Gasket encompasses
the use of Pascal's triangle. To create the Sierpinski Gasket using Pascal's Triangle,
every odd number in the progression of the illustration shown, must be included in the
gasket, while every even number will not be included.
A fractal that is closely related to the Sierpinski Gasket is called the Sierpinski Carpet.
The Sierpinski Carpet is a very important fractal when dealing with antennae. This fractal
is used in technology such as cell phones, tablets, high-tech watches and computers.
It is used to organize the antennae as well as allowing for additional signals to be received.
It is important to realize that in addition to using fractals to create more efficient
antennae, fractals were being founded in other places in the 1980's. For instance, at IBM
disruptive frequencies were preventing the transmission of computer data over telephone
lines. The frequencies were simulated and mathematicians noticed that within these frequencies
there were smaller frequencies of similar form, and of course self-similarity meant
that fractals must be present. Global warming is an increase in the earth's average atmospheric
temperature that causes corresponding changes in climate, which may result from the greenhouse
effect. The greenhouse effect is an atmospheric heating phenomenon, caused by short-wave solar
radiation being readily transmitted inward through the earth's atmosphere but longer-wavelength
heat radiation less readily transmitted outward, owing to its absorption by atmospheric carbon
dioxide, water vapor, methane, and other gases. Basically, rising levels of carbon dioxide
are causing concern. This is where trees play an important role. Trees, as well as other
plants, absorb carbon dioxide for the process of photosynthesis. In other words, the more
trees there are the less carbon dioxide in the air and the less global warming will occur.
Using fractal dimensions to describe trees and further entire forest ecosystems, may
assist us in understanding the amount of carbon dioxide being absorbed throughout the world
and the number of plants that need to be grown in order to put a stop in global warming,
or at the very least, to slow down the global warming process. The fractal dimensions help
because instead of measuring every tree and branch and twig in a forest, a survey of one
tree's measurements can be used to calculate the carbon dioxide of the entire forest. To
put it simply, one tree would be measured for the amount of carbon dioxide it would
take in. From this tree, there is a distribution of different sized branches. This distribution
should match the distribution of sizes of trees in the entire forest, meaning the forest
is one enormous fractal itself.Benoit Mandelbrot was Polish-born on November 20, 1924 and was
a French Mathematician. Since he was the catalyst for the discovery of fractals, he is best
known as the Father of Fractal Geometry. He is credited with coining the name fractals
to describe objects and surfaces, which are irregular at various dimensions of scale.
It wasn't until 1975 that Mandelbrot's fractal geometry replaced Euclidian geometry, which
had dominated our mathematical thinking for thousands of years. We now know that Euclidian
geometry pertained only to the artificial realities of the first, second and third dimensions.
These dimensions are imaginary. Only the fourth dimension, which is the dimension of the space-time
continuum we live in, is real. Mandelbrot asked an excellent question: how can one determine
the length of a coastline without following the entire shore with measuring equipment?
This question, as well as other mathematical problems, began Mandelbrot's search. His
search ultimately lead to his studies on fractal dimensions, where he introduced a way of measuring
the degree of roughness or irregularity of an object. Although Mandelbrot should be credited
with the majority of the study on fractals, it was Gaston Julia's modern dynamical systems
theory that saved Mandelbrot from hours upon hours of advanced calculations. As a result,
Mandelbrot relied on computers to plot images and calculate the seemingly endless iterations.
Without this technology, it is unlikely Mandelbrot would have been able to discover as much as
he did. One unique and very famous observation made
by Mandelbrot is known as the Mandelbrot Set. This famous quadratic recurrence equation,
z = z^2 + c , is another example of a design with complex irregularities, that is self-similar
in itself, with a base shape that is similar to the image on the left below. The resulting
self-similar iteration can be as complex as the image on the right below.
Shown here are others who greatly helped with the exploration of Fractals and their contributions.
Fractals can also be found in the medical field. Application of fractal geometry to
cell biology stemmed from the discovery that cellular membrane systems had fractal properties.
Areas such as stem cell research, radiation treatment, and other uncertain cures may all
be ruled out if fractals take a bigger role in the field of medicine. The respiratory,
circulatory, and nervous systems are great examples of branches subdividing over and
over in a fractal form. Below is a cast of lungs that clearly show the symmetry or self-similarity
that every ultra-organized fractal has. It was shown that normal vasculature has a fractal
dimension of 1.7 while tumours have a fractal dimension of 1.89. Meaning, tumours are a
more chaotic vasculature, making them stand out from surrounding vasculature.Charles Darwin
was a Biologist and Scientist born February 12, 1809 in Shrewsbury, England. He is best
known for his work as a naturalist, developing the theory of evolution and the process of
natural selection. However, the question of "design" in nature troubled Darwin all
his professional life. Darwin once wrote, "I am conscious that I am in an utterly
hopeless muddle. I cannot think that the world, as we see it, is the result of chance; and
yet I cannot look at each separate thing as the result of design." Many other scientists
explained the "design" of nature as just occurring because of the creator of this world
- or God. It was said that the perfection of nature was due to arbitrariness and accident.
Darwin, however, seems to have been correct in his assumption that the world cannot be
created by accident. As mentioned before, self-similarity can be seen in many aspects
of nature, from rivers to mountains to clouds to seashells. All these aspects of nature
seem to be in chaotic forms but fractal dimensions can often describe their immense irregularities.
Fractals created a reason for design in nature, thus solving the problem Darwin had with natures
perfection. As in a shoreline, lightning's fractal dimension
is spectacularly arbitrary and irregular. Observing the main stem of electricity emanating
from the lightning, it appears to be similar to a tree trunk. Lightning can be compared
very closely to the irregularity of a tree but at a greater fractal dimension. Peacocks
are great examples of fractals in the animal kingdom. Of course, the fractals are in the
tail of these creatures. Also shown is a very famous fractal of a seashell. The perfect
pattern of the seashell is called the Fibonacci Spiral. Delicate Queen Anne's Lace also
demonstrates a fractal. This plant is really just wild carrot. It is a good example of
a floral fractal, as each blossom produces smaller iterative blooms. The image shown
was shot from underneath the plant so the iterations can be easily observed. Lastly,
there is a special type of broccoli. It is cousin to the cabbage and is a particularly
symmetrical fractal. This broccoli is called Romanescu. Of course, in addition to these
examples, there are innumerable fractals found in nature. If one were to think of the human
body, aren't arms and legs fractals also? The individual phalange of the finger separated
by joints could be considered iterations of the larger phalange before it and the largest
phalange could be seen as the iteration of the arm. This can also be observed with legs
and the toe phalanges. Paul Cezanne, a painter born January 19, 1839 in Aix-en-Provence,
France, once wrote, "Everything in nature is modeled according to the sphere, the cone,
and the cylinder. You have to learn to paint with reference to these simple shapes…"
However, when Benoit spoke of Fractals, he took a different approach by saying, "Clouds
are not spheres, mountains are not cones, coastlines are not circles, and bark is not
smooth, nor does lightning travel in a straight line." Since, the arts are often mirrored
after nature, it would not be surprising to see fractals in the art itself. Even if we
are unconscious of this fact, they are there. Also related to Fractals is computer generated
art, specifically artwork created using fractals themselves, have been made by uninformed programmers,
who use Mandelbrot's Set (z = z^2 + c) with a z value other than 0. These artworks turned
out to be hideous and are referred to as Mandelbrot Roadkill.
An artist named Jackson Pollock, born January 28, 1912 in Cody, Wyoming, unconsciously adopted
the language of nature, or fractals, to create his own artwork in a very self-similar form.
Pollock began with a canvas lying in a horizontal position, and used a continuous stream of
paint and gravity to achieve his painting. When compared to paintings such as Pollock's,
where drip trajectories were studied, the results were strikingly similar. The painting
on the left demonstrates a pendulum freely swinging. The paint lines are in a non-chaotic
form. However, the middle painting was achieved when the pendulum was knocked at a frequency
slightly lower than the one at which it naturally swings. When examining a smaller portion of
these paintings, it can be seen, that a small section has similar statistical data compared
to the entire canvas. If a person gleaned only one piece of knowledge from this paper,
it should be the fact that fractals are everywhere. Fractals make up the food we eat, the flowers
we keep in our homes, the various parts of our bodies, the artwork we observe, the décor
we use at home, the weather just outside our doors, and even the animals we keep. When
so many things in this world seem so random and chaotic, fractals make sense of the arbitrariness.
This topic is extremely new and very under-explored. More attention needs to be paid to the world
of fractals, more importance needs to be placed on learning about fractals, and more studies
need to be completed about fractals. This topic of discussion is not just some cool
mathematical element, but instead, blends mathematics with every sphere of knowledge.
From nature, to art, to medicine and beyond, fractals are showing themselves and giving
the world a new avenue to explore and from which to gain great insight. Exploring fractal
dimensions could aid in finding cures for cancer, find sanity in seemingly random paintings,
and give lucidity to beautiful aspects of nature. Clearly, this is the perfect topic
to delve into for the betterment of human understanding.
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