Many of us think that clouds are spheres, mountains are
cones, coastlines are circular and lightening travels in a straight line. Well,
what if I told you this was not the case? Benoit B. Mandelbrot, a Polish-born,
French and American mathematician also attests to this claim in his book, “The
Fractal Geometry of Nature” which I used as one of the references in this
essay. He refers to these natural forms and many other man-made creations of
this kind as “Rough”. And to study this roughness, which he also refers to as “irregularity”,
he invented the term “Fractal”, and he devised a new kind of mathematics called
Fractal Geometry to study these irregular shapes.
A fractal is described as a never ending pattern that
repeats itself on different scales. This behavior is referred to as
“self-similarity”. Take, for example, broccoli. If you remove one branch from
the whole, the branch looks like whole broccoli with smaller branches. You can
then take another branch from the branch that you initially took from the whole
and it will look like the whole broccoli and the bigger branch but smaller in
size. This process can go on and on and on until you reach the innermost parts of the broccoli. This is a fractal. A fractal is made by repeating a
simple process again and again. Fractals can be very complex, and in most cases
infinitely complex; for instance, one can zoom in on a shape and find the same
shapes again and again.
As aforementioned, fractals occur naturally and some are
man-made. They are found in Nature, In Geometry and In Algebra.
In nature, fractals are found in the form of branching and
spirals. From neurons in our bodies, the structure of the lungs to trees and
lightning. The branching of these objects creates fractals; for example, take a
tree, a sprout branches, then the branches branch and those branches in turn
branch and the process goes on and on. It is the same in the other objects that
result in fractal branching.
Fig. 1: The lung as an example of branching fractals
On the other hand, Spirals which are other common fractals in nature, are found in plants such as
agave cactus, sunflower, pinecones etc. It is also exhibited on a large scale
in hurricanes, galaxies etc.
Fig.2: An agave cactus as an example of spiraling fractal.
In Geometry, I will use the Sierpinski Triangle and The Koch
Curve to depict how fractals occur. The Sierpinski Triangle is made by
repeatedly removing the middle triangle from the previous generation (remaining
triangles). The number of black triangles increases by a factor of 3 in each
step; 1,3,9,27,81,243,729,…,n (as illustrated in the figure below)
Fig.3: The Evolution of the Sierpinski Triangle
The number of triangles in the Sierpinski Triangle can be
calculated by the formula: N=3k -1, where N is the number of
triangles and K is the number of repetitions. For every repetition, the area of
the Sierpinski Triangle decreases by a factor of ¼. So, a question to chew on as you continue with
the reading; what happens to the area of the triangle as the number of
repetitions approaches infinity?
The Koch curve, on the other hand, is a curve made by
repeatedly replacing each segment of the generator shape with the smaller copy
of that generator. It is constructed as follows:
Step 1: Draw a line segment and divide it into three equal
segments
Step 2: Draw an equilateral triangle whose base is the
middle segment of the three segments from step 1
Step 3: Remove the base of the triangle drawn in step 2
The development of the Koch curve is illustrated in the
figure below
Fig. 4: The evolution of the Koch Curve
The curve begins to look like a coastline with more repetitions. The total length of the curve increases by one third with
each repetition, and each repetition results in four times as many segments as
the previous repetition, and the length of each segment is one third the length
of the segments in the previous repetition. Now, taking n to be the number of
repetitions, and taking into consideration the original perimeter of the generator triangle, can you derive a formula that can be used to express the length of
the curve after n repetitions? What happens to the length of the curve as n
goes to infinity?
Fractals can also be
created by repeatedly calculating a simple equation again and again. As the
equation gets bigger and approaches millions, the use of a computer becomes
inevitable. This brings in the Mandelbrot set, which was discovered by Benoit
B. Mandelbrot in 1980, shortly after the invention of the personal computer. The set has the following equation:
Zn+1 =Zn2
+ C
It works as follows: Plug in
a complex number “C” into the equation above. The equation gives the new Z
denoted as (Zn+1) in the equation. Plug this value back into the equation
as the old Z (Zn) in the equation and calculate it again. Do the
process repeatedly for as many times. Note that each complex number is a point in a 2-dimensional plane. For most values of ‘C’, when you square them,
they get bigger and bigger and tend to infinity. However, not all values behave
in this way; instead some values get smaller, or alternate between a set of
fixed values. This defines the Mandelbrot set. It is a set of complex numbers ‘C’ for which
the sequence (C, C2 +C, (C2 +C) 2+ C, ((C2+C)2+C)2
+C,…) does not approach infinity. For example, take -1 and plug it into the above equation; You should see that the iterations are bounded between -1 and 0 and
do not increase. The illustration below
depicts a set of complex numbers which satisfy the Mandelbrot set.
Fig. 5 The Mandelbrot set on the plane
Interestingly, Fractals are not only restricted to
Mathematics, but can also be observed in curves of financial markets. Fractal
jumps can be observed in graphs of price increases over time. The figure below
shows a model of price changes over time. The graph conforms to the Brownian
motion, which is the mathematical model used to describe such random movements
as price changes overtime.
Fig. 6: The Fractal behavior of Prices
In the end, why bother to study fractals or even recognize
their existence? Well, besides that fractals are beautiful patterns and fun to
make, they also have many uses; for example, computer chip cooling circuits
have a fractal branching pattern which enable them to channel liquid nitrogen across
the surface to keep the chip cool, Fractal antennas used in cellphones and
other devices, and in the health sector, researchers use fractal analysis to assess
the health of blood vessels in cancerous tumors. I picked this topic out of curiosity,
but, which later on changed into interest. I have observed fractal behavior throughout my
life but never cared for its importance, but thanks to Benoit B. Mandelbrot for
enlightening me and many others on this concept.
REFERENCES:
1.
"Fractals and the Art of Roughness." Benoit
Mandelbrot:. N.p., n.d. Web. 06 Nov. 2014:
2.
10.01.08, Posted. "A Radical Mind." PBS.
PBS, 09 Jan. 0000. Web. 08 Nov. 2014
3.
"The Mandelbrot Set - Numberphile." YouTube.
YouTube, n.d. Web. 08 Nov. 2014.
4.
Mandelbrot, Benoit B. The Fractal Geometry
of Nature. San Francisco: W.H. Freeman, 1982. Print
5.
Mandelbrot, Benoît, and Richard L. Hudson. The
(mis)behavior of Markets: A Fractal View of Financial Turbulence. New York:
Basic, 2008. Print.
Images: http://fractalfoundation.org/
Images: http://fractalfoundation.org/
I also researched about fractals, so when I was reading this and still found new things that I didn't know was pretty cool. For example, how our lung is structured like a fractal. It's astonishing to see how many things can be based on this concept, whether it is in nature or is man-made. Fractals may seem like a superfluous topic to research about, but it does yield helpful results like, from what I found, creating beautiful architect and stronger radio signals.
ReplyDeleteI was really amazed to learn that fractals can be found in a lot of things in life and nature. Reading this blog about the fractals and the mathematics behind it made me want to look more into reading about fractals. It's very cool to discover the different fractal patterns as well.
ReplyDeleteI never knew that fractals could be found in so many places! The ones that especially caught my eye were those in the lungs. I am a biology major so that kind of thing is really cool to me. I read Zach's post before your's but I definitely learned some more things about fractals!
ReplyDeleteThe concept of fractals is very interesting as you pointed out how it is seen in many things in life. It's also used to analyze patterns in the fields of business, medicine, and mathematics, as well as others. As Zach mentioned above in the comments, they are useful for those areas of architect and radio signals.
ReplyDeleteThe concept of fractals and the patterns fractals produce are very new and interesting to me. I enjoyed the picture examples you put in your summary, they helped with your explanation of the pattern and better defined what was going on. I particularly enjoyed the spiral fractal because of the many examples it exhibits in nature. Great Summary!
ReplyDeleteThis is an interesting topic which has many different illustrated examples to help me understand this topic. The different pictures make the different concepts easy to understand. Overall I thought this was a great summary with great examples.
ReplyDeleteFractals have always been one of my most favorite mathematical concepts. They are infinitely beautiful and the fact that some are naturally occurring is amazing. i was not aware of the formula that could be used to create them though. You sir have stimulated my mind. thank you.
ReplyDeleteThe concept of fractals is very interesting, especially how you explained them. You explained them well. It's amazing that fractals can be in many different places.
ReplyDeleteThe only knowledge I have of fractals is reading the other blog post. The big difference between this one and the other post was the formula you put in. I thought that part was really interesting. It's tough to make a good comment after just righting one about fractals. I'm sorry I didn't read yours first.
ReplyDeleteFractals, as mentioned before, are really interesting to look at in a mathematical and illustration kind of aspect. I liked how the Koch Curve and the Fractals are somewhat similar, especially in the way they multiply.
ReplyDeleteOne of my favorite things is the occurrences of patterns in nature. It's really cool to think about the structure of the world as we know it. One of my favorite things is to take things that some people see as useless and seeing the applications that can used.
ReplyDeleteI liked the picture examples you used in your summary with the fractals. I also liked how you included the formulas with the fractals. I also liked the examples you gave such as the lungs being an example of a fractal.
ReplyDeleteI never knew our lung was shaped like a fractal. I thought that was interesting to know. I also liked how you included those pictures in the blog. They were interesting photos as well. It's also crazy that the fractals can be in so many places.
ReplyDeleteThe fractal was explained in great detail here, and I the examples given were amazing. Along with others, it is unbelievable that the lung is shaped like a fractal. Even though this was previously known, the realization and the picture is truly eye opening. The example of the neurons was also really interesting. Fractals are everywhere and interesting.
ReplyDelete