Fractals: Mathematical Beauty
A Brief Exploration
by Alex Sredenschek
Table of Contents
p. 2.....................................................................Foreword
p. 4.........................................................Defining Fractals
p. 8...................................................................Dimension
p. 10.........................Hausdorff-Besicovitch Dimension
p. 14...............................................................Koch Curve
p. 18..................................................................Cantor Set
p. 20..........................................................Mandelbrot Set
p. 28......................................................................Julia Set
p. 38..............................................................Applications
p. 41..............................................................Works Cited
Foreword
The purpose of this book is to give a glimpse
into the complex, yet fascinating nature of
fractals. It is intended for primarily those who
do not have a strong background in higher-level
mathematics, but for those who wish to learn
about the exciting nature of fractals. To do this,
there are visual examples throughout the book
as well as general explanations of a subject that
has filled countless textbooks and has been
studied by mathematicians for centuries. In this
book are several classical fractals as well as some
of their properties. At the end are some
applications of fractals and their representation
in nature.
Read for knowledge, curiosity, and,
of course, fun.
Fractals: Definition
Simply put, a fractal is a mathematical set or a natural
phenomenon that repeats itself on all scales. One can also say
that it is an odd geometric shape, though still exhibiting a
pattern. Therefore, they exhibit some form of similarity and
repeat themselves in patterns (Clayton n.pag.).
"I coined the term fractal from the Latin adjective fractus. The
corresponding Latin verb frangere means "to break:" to create
irregular fragments. It is therefore sensible--and how
appropriate for our needs!--that, in addition to
"fragmented" (as in fraction or refraction), fractus should also
mean "irregular," both meanings being preserved in fragment
(Mandelbrot 4)."
New computer technologies have helped the discovery of
many new fractals and their properties as well as older
ones.vomputers allow for much deeper exploration, allowing
for not only mathematical discovery but for increased
aesthetic beauty.
A fractal generated by advanced software (Frax)
using 3-D visual effects.
Dimension
Dimension in mathematics is incredibly important. On the
surface, it may seem very simple. Take a line, for example. It
has a dimension one because there is only one coordinate to
describe a location on the line. With a square, the dimension is
two with the same logic: there are two coordinates required.
Finally, with the cube, there are three coordinates required,
and therefore, it has three dimensions. Dimension, then, can
be defined as the number of coordinates required to specify
the location of a point on a shape or object (Weisstein n.pag.).
So, therefore, it seems logical that dimensions can only take
on integer values.
However, this is false.
Standard, or Euclidian, geometry deals with dimensions of
integers or whole numbers which apply to regular geometric
shapes. Fractals, on the other hand, have different properties
than shapes like squares, cubes, or triangles.
Hausdorff-Besicovitch Dimension
As I said before, fractals can have dimensions
other than integers, and this dimension is
known as the Hausdorff-Besicovitch Dimension
or, simply, the fractal dimension. They were
studied by German mathematician Felix
Hausdorff, and he introduced them around 1918
(Clayton n.pag.). What these fractal structures
have, then, are properties of the integer
dimensions that surround them. A fractal with
dimension 1.2 would have properties of both a
one-dimensional straight line and a twodimensional shape. This is largely a reason why
fractals are so different from standard geometric
shapes.
Calculating the Hausdorff-Besicovitch
Dimension
Calculating the fractal dimension comes from the expression,
N=r^D
and it also applies to standard geometric shapes (Clayton n.pag.).
The variables are related as follows:
Where N denotes the number of similar images that result with
each iteration of the pattern,
r denotes the magnification factor,
and D denotes the dimension of the object.
So, with a little algebra, solving for D yields:
D=ln(N)/ln(r)
Let's look at a straight line and a square as examples.
Take a straight line of dimension D=1 and divide into parts of
equal length. Let's say we want to make a line divided into 3
parts. The length, then, of each line segment would be 1/3. If
we want to divide it into 5 parts, the length of each line
segment would be 1/5.
We can generalize this into the expression 1/r, where r denotes
the magnification factor as stated earlier. Substituting values
into D=ln(N)/ln(r) yields a dimension D=1.
Now, look at a square. If we divide both sides by some integer
r, we will have N number of equal squares with each iteration.
If we divide the square's sides by two with each iteration,
there are 4 equal squares after the first, 16 after the second, 64
after the third, and so on. Substituting values into D=ln(N)/ln
(r) yields a dimension of D=2
So, how could a shape have a dimension other than an integer?
Let's look at some examples of well-known fractals.
The Koch Curve
Swedish mathematician Niels Fabian Helge von Koch
constructed a fractal that is now named after him: the
Koch Curve, also known as the Koch Snowflake. To
create this fractal, we take a single equilatreral triangle
and extend an equilateral triangle from the middle.
Then, we delete the base of the triangle, leaving us
with four line segments (for each side) of 1/3 the
length of the initial line. The process is then repeated,
on each of these four line segments, creating 16
segments of length 1/9 the original length (Ward
n.pag.).
When this process is reiterated an infinite number of
times, the Koch Curve results. The dimension,
however, is not an integer value.
To calculate the dimension of the Koch Curve,
we must find N and r. With every iteration, four
new segments are created, each of length 1/3.
Thus, the equation N=r^D becomes: 4=3^D
Therefore, the dimension of the Koch Curve is,
D=ln(4)/ln(3)~1.262
which is clearly not an integer.
The Koch Curve under many iterations with
computer software gains increasing detail, making the
fractal even more intricate. If one were to zoom into
the Koch Curve, one would see self-similarity, because
the sides of the shape are identical, but on different
scales, a defining property of fractals (Clayton n.pag.).
Something else that is also special about the side of the
Koch Curve is that it is infinite in length. After the
first iteration, there are 4 segments of length 1/3, and
after the second, 16 segments of length 1/9, and so on,
modeled by (4/3)^n where n is the number of the
iteration. As n approaces infinity, however, the length
of the curve does, too, because 4^n > 3^n (Clayton
n.pag.).
The Cantor Set
In the late 19th century, German mathematician
Georg Cantor created a fractal known as the
Cantor Set (Shaver n.pag.). To create it, take a
straight line, and with each iteration, remove the
middle third of each line (Surgent n.pag.).
Similar to the Koch Curve, it also has a noninteger dimension. Looking at just one of the
two line segments produced with each iteration,
it is one third less than the original segment, so
r=1/3. Also, there are half as many segments,
when just looking at one of the two, so N=1/2.
Substituting these values into D=ln(N)/ln(r)
yields D~0.6309
(Surgent n.pag.).
A Cantor Set shown up to its
fifth iteration.
The Mandelbrot Set
Mathematician Benoit B. Mandelbrot
intensively studied fractals, and the Mandelbrot
Set is named after him. It is a set of complex
numbers, which means it includes both real and
imaginary numbers (Devaney n.pag.). An
example of some complex numbers are:
3+4i and 0.5-1.2i where i^2=-1
To the right is the Mandelbrot Set. The region
colored in black are all of the numbers such that
the equation of the set does not diverge to
infinity. Thus, the values of c in the white space
do diverge to infinity and are not included in the
Mandelbrot Set (Devaney n.pag.).
The Mandelbrot Set
The equation for the Mandelbrot Set, a
quadratic recurrence equation, is,
with the condition that the initial z value is
equal to zero, and where c denotes some
complex number (Williams n.pag.).
To get from this equation
to the image on page 21, it
is critical to understand
the definition of the
Mandelbrot Set. It is the
set of numbers such that
the sequence does not
diverge to infinity as
infinitely many iterations
are carried out (Devaney
n.pag.). For example, let
c=2.
Clearly, the sequence
continues to get bigger
and bigger with each
iteration. Therefore, the
value c=2 is not in the
Mandelbrot Set.
Look at another example.
Let c=-1.
Instead of getting
increasingly large, the
sequence alternates
between 0 and -1. It does
not get very large.
Therefore, the point c=-1
is within the Mandelbrot
Set. In fact, this is evident
in looking at the domain
of real numbers for the
Mandelbrot Set, equal to
the closed interval [-2,
0.25].
On another note, the Mandelbrot set is not
exactly self-similar, but it certainly does have
repeating patterns throughout it. Take, for
example, these images:
Zoomed in from the far right
portion of the Mandelbrot
Set. It has entirely different
patterns than the image on
page 21.
Zoomed in on a center spiral
of the image to the left.
Notice how even more very
similar patterns appear on
smaller scales.
The Mandelbrot Set on page 21, but colored and textured. These
many different colors and effects allow for even greater aesthetic
appeal of fractals.
A region of the Mandelbrot Set zoomed with a different color
scheme and texturization than the image on the left.
Julia Set
Julia sets are named after French mathematician
Gaston Julia, born in 1893 (Frazer n.pag.). He first
brought forth the idea of Julia Sets in 1918 in an essay
of his (n.pag.). Interestingly, the Julia Set not only has
the same equation as the Mandelbrot Set (see page 22),
but they are inter-related. The one relationship
between the two is that the c that is variable for the
Mandelbrot Set is fixed or held constant in the Julia
Set. The value that changes is z, a complex number
(Williams n.pag.).
In fact, the constant c becomes the center of what is
called the Filled Julia Set. (The Julia Set is simply the
boundary of the Filled Julia Set). Every point c in the
Mandelbrot Set has a unique Filled Julia Set that
corresponds to it (Devaney n.pag.).
Filled Julia Sets, like the one
above, are striking in
appearance, exhibiting
intricate patterns.
Thanks to fractal-generating
software, colors and visual
effects can enhance the beauty
of these fractals.
Like the Mandelbrot Set, the Julia Set is
concerned, primarily, with variables that make
the sequence diverge to infinity, or converge to,
approach, or oscillate around a value.
This is how coloring works with fractal-generating software.
Some values of z take much longer to diverge, if they do at all.
Values that do not diverge are usually colored black, but they
can be colored any color, like the Filled Julia Set below.
Certain colors are assigned to the number of
iterations it takes for a certain value z to make
the sequence diverge to infinity. These points
are colored accordingly (Williams n.pag.).
The most intriguing property of the Mandelbrot
Set and Filled Julia Sets is their close
relationship. The Mandelbrot Set, itself, is
theoretically infinite. So, if there are an infinite
number of points c within the Mandelbrot Set,
there are also an infinite number of Filled Julia
Sets within the Mandelbrot Set. But it doesn't
stop there, because each Filled Julia Set is also
theoretically infinite.
Infinity within infinity.
This relationship is just one of the many
incredible phenomena in mathematics as well as
one of many examples of the mysteries of
infinity.
Above are the axes for the
Mandelbrot Set, with the
horizontal axis containing real
numbers and the vertical axis
containing imaginary numbers.
To the right are some Filled
Julia Sets and their c values.
Using the above image, examine
how the Filled Julia Sets change
with their c values.
The far left end
(straight line)
c=-2
Near middle of
upper circle
c=-0.1+-.75i
The far right end
c=0.25
Inside boundary of
upper circle
c=-0.18+0.68i
Inside of large
circle to the left
c=-1
Outside of boundary
of upper circle
c=-0.18+0.66i
Within boundary of upper circle to the left
c=-0.783+0.095i
Outside of boundary of upper circle to
the left
c=-0.783+0.195
Far left
c=-1.7557
Perfect circle of radius 1
c=0
Applications
Fractals are complicated. They aren't regular, geometric shapes.
However, this is why they are so applicable to the universe we live
in. Mandelbrot says it very eloquently,
"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" (Mandelbrot 1).
It is inaccurate to generalize the breathtaking phenomena that
surround us, from lightning to plants to rivers and more.
In Biology, research has been done on the presence of fractals in
rainforests. It was proposed by Geoffrey B. West, James B. Brown,
and Brian J. Enquist that the stucture of a rainforest is closely
related to the structure of its individual trees. By studying just one
tree's relationships between its branches and its trunk, streams of
information are available, most notably the carbon dioxide
absorption of a rainforest (Schwarz, Fractals n.pag.)
When dealing with technological devices that require the use of
various frequencies such as telecommunications, fractals are
invaluable. The self-similiarity of the antenna can effectively
optimize its range of accepted frequencies. At the same time, the
fractal structure of the antenna allows for much smaller hardware,
allowing for smaller devices with more properties.
A healthy hearbeat, too, exhibits fractal patterns. Cardiologist Ary
Goldberger in Boston found that the patterns of healthy hearbeats
were rather irregular. However, upon magnifying graphs, the
irregularities showed somewhat self-similar patterns that strikingly
resembled mountain ranges and their jagged appearance. Because
of this fractal blueprint of healthy hearbeats, doctors can use it to
detect problems with a patient's heart early on.
Fractals are also very useful in special effects. In Star Wars III:
Revenge of the Sith, to make the lava spouts near the end of the
film, Willi Geiger, a special effects designer, used a fractal pattern to
make the lava seem much more realistic. He continuously
reiterated the patterns of a swirl on smaller and smaller levels,
creating a very realistic lava splash (Schwarz, Fractals n.pag.)
Mathematics has a whole new dimension to
explore with the faculties of modern technology
and fractals. The subject is ever-growing with
new discoveries as mathematicians dive into the
mysteries of the universe. Yet, on the way, they
encounter wonders, some of which connected in
some way. I marvel at the similarity between
lightning, tree roots, and human veins, even
though they are so different. Fractals are even
inside of us!
Mathematics never fails to find some way to
both amaze and confuse. There is so much
already known, but compared to the many
enigmas of the universe, it is nothing.
Humans are curious and persistent beings,
though; discovery awaits at every step.
Works Cited
Clayton, Keith. "Fractals & the Fractal Dimension." Basic Concepts in Nonlinear
Dynamics and Chaos. N.p., 28 June 1998. Web. 01 June 2015. <http://
www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html>.
Devaney, Robert L. "Unveiling the Mandelbrot Set." Plus. N.p., 01 Sept. 2006. Web. 01
June 2015. <https://plus.maths.org/content/unveiling-mandelbrot-set>.
Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W.H.
Freeman, 1983. Print.
Ferguson, Stephen C. Tiera-Zon. Computer software. Tiera-Zon. Vers.
2.7. N.p., 9 Dec. 1997. Web. 31 May 2015. <http://1998.tierazon.com
Tierazon/Tierazon.html>.
Fractals: Hunting the Hidden Dimension. Dir. Michael Schwarz and Bill Jersey. PBS,
2008. DVD.
Fraser, Jonathan. "Introduction to Computability." Algorithms and
Computation in Mathematics Computability of Julia Sets (2009): 1-19. 6
Apr. 2009. Web. 02 June 2015. <http://www.gvp.cz/~vinkle/mafynet/
GeoGebra/matematika/fraktaly/linearni_system/julia.pdf>.
Shaver, Christopher. "An Exploration of the Cantor Set." (n.d.): n. pag. The Electronic
Proceedings of the Missouri MAA. 2009. Web. 02 June 2015.
Surgent, Scott. "The Cantor Set and Fractal Dimension." N.p., n.d. Web. 01 June
2015. <https://math.la.asu.edu/~surgent/mat271/cantorset.pdf>.
Ward, Matthew. "An Introduction to Fractals." An Introduction to
Fractals. N.p., n.d. Web. 01 June 2015. <http://davis.wpi.edu/
~matt/courses/fractals/intro.html>.
Weisstein, Eric W. "Dimension." MathWorld--A Wolfram Web Resource.
Wolfram Alpha http://mathworld.wolfram.com/Dimension.html
Williams, Alun. "The Mandelbrot Set and Julia Sets." The Mandelbrot
Set and Julia Sets. N.p., n.d. Web. 02 June 2015. <http://
www.alunw.freeuk.com/mandelbrotroom.html>.
Weiss, Ben, Kai Krause, and Tom Beddard. Frax--The First Realtime
Immersive Fractals. Computer software. Apple App Store.
Vers. 1.31. Iter9, LLC, n.d. Web. 02 June 2015.
The images on the front and back covers, and pages 3, 5, 6, 7, 21, 25, 26, 27, 29, 30, 31,
33, 34, and 37 were made with the help of the iOS application Frax--The First Realtime
Immersive Fractals.
The images on pages 35 and 36 were made with the help of the computer program
Tiera-Zon.
I want to deeply thank Mrs. Scherer,
my calculus teacher, not only for the
materials and help she gave me in
researching fractals, but for inspiring
in me a strong desire to learn more
about the fascinating, challenging,
and rewarding field of mathematics.