Papapietro 1
Fractals
Jacqi Papapietro
Extra-Credit Assignment
Math 251 Section 4
This morning while getting my caffeine fix at the hub I asked my music-major roommate to read over my paper. This is how the scene went:
“Hey Christine, will you proof-read my paper about fractal geometry for math class?”
“Well…okay, but…fractal geometry??…just sounds like more mathematical concepts and crazy equations to learn. Why do we need yet another kind of geometry? Can’t we just stick with squares and circles??”
In the simplest way possible, I explained it to her “Lay this piece of paper down on a table. The paper lays in two dimensions, the x and y planes. Now take the paper and crumple it into a ball (keeping in mind that you will have to flatten it out again to finish reading my report). The paper is three dimensional while in a ball. Okay, now flatten it again and lay it back on the table. Now what dimension is the paper? The paper is not completely flat, but not completely balled-up either. Using Euclidean geometry, it must be labeled with a dimension of an integer. So is the half-crumpled paper two- or three-dimensional? It is neither. This is where fractal geometry takes center stage. Using techniques in fractal geometry we can figure out dimensions that are fractions!”
“Okay, I get that. But why do I care what dimension your paper is in?” she replied.
“Fractals relate to so many things, like plants, weather, fluid flow, geologic activity, planetary orbits, human body rhythms, animal group behavior, and socioeconomic patterns, just to name a few. And here is something right up your alley. Fractal geometry can even be used to make music (1)! So will you please read my paper? You can see what all this fractal stuff is about.”
What is a Fractal?
Webster’s dictionary defines a fractal as “any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size.” Mandelbrot, the “father of fractals” defined a fractal as“a set whose Hausdorff dimension is not an integer.” In math terms, a fractal is a geometric shape that is self-similar and that has fractional dimension. For a fractal to be self-similar, it must appear the same shape an every scale. Having fractional dimension simply means that the dimension doesn’t have to be an integer, but can be a fraction (1). Here are some pictures of fractals.
(2)
The first two graphics are of a fern. It is important to note that in order for something to be self-similar, it is not necessary for the shape to be exactly the same at every scale, only the same type of shape (2). You may have noticed that the second two lines of graphics appear quite different than the last line. This is because they were created using different methods.
Creating a Fractal
Perhaps the most famous fractal is Sierpinski’s Triangle. To make this triangle using the first method, start with an equilateral triangle. Connect the midpoint of each side of the triangle and “cut out” the center triangle that is made.
(3)
Keep repeating this process and you will end up with Sierpinski’s Triangle. As you can
(3)
In the second method of creating a fractal, instead of removing or “cutting out” sections of the shape, sections are added to the original shape. The following graphic is of the Koch curve.
(4)
These two methods work similarly, in that you always add or subtract lines or shapes to every original shape, and you always keep repeating the same steps. Below we can see how the Sierpinski Carpet can be generated using each method. The result is the same.
Using the “cut-out” method:
(1)
Using the method of addition:
Finally, (1)
Calculating the Dimensions of Fractals
As seen earlier by crumpling and un-crumpling my paper, we found that the resulting dimension of the paper was somewhere between the integers 2 and 3. The more crumpled and bumpy the paper is, the closer it is to being three-dimensional. The more it is smoothed it out, the closer it is to two-dimensional. The same applies in nature. When analyzing a coastline using fractals, the more curved or bumpy the coastline, the more it is like a two-dimensional object and the more straight the coastline, the more it is like one-dimension. But how do we go about calculating a fractal dimension?
Let’s take some simple examples with integer dimensions. We start with a line:
(1) This simple line can be considered a fractal because it is self-similar. When the line is divided into four segments, each of the segments looks the same as the original line. In this picture, the scaling factor is 4, because when each small segment it enlarged by a factor of 4, it is exactly the same size as the original line. Since there are four resulting segments (N = number of resulting segments, shapes, or portions), and the scaling factor is 4 (S = scaling factor), we can come up with the equation:
4 = 41, orN = SD, where D = dimension
We know that a line is one dimensional, so in this instance, the equation holds true. We can also use the equation for objects in dimensions great than 1, such as a cube.
(1)
The equation holds true because we know that this cube is, in fact, three dimensional.
We can also use the equation N = SDto calculate fractal dimensions.
Although we can use this equation to find a fractal dimension, it is not as easy as reading the exponent in the equation, (as in integer dimensions). For instance, let’s look at Sierpinski’s Triangle.
(1)
In order to find the dimension, D, we must use logarithmic properties:
3 = 2D
log3 = log (2D) → log3 = Dlog2
log3/log2 = D
D = 1.58496
So, the equation, N = SDcan be rearranged to log N/log S = D, and used to find fractional dimensions. With this new method of calculating dimension, it is possible to find the dimensions of objects, such as a mountain, cloud, or half-crumpled sheet of paper, that cannot be described with integer dimensions (1).
Using fractal geometry, things such as area and volume can also be calculated for fractals. However, fractal dimensions are very important. Many studies are being conducted on their relevance and/or usefulness to nature and music.
Fractals in Nature
Self-similarity, a property of fractals, is seen frequently in nature. Recently, self-similarity in the African plant Asparagus plumosus has been found, identifying the plant as a fractal.
(5)
In this specific study, scientists used the box counting method. In this method, a sample (a branch in this case) is placed on a grid. Every square that the sample lays even partially in is shaded and counted. To calculate the fractal dimension (D) of the sample, you must
(5)
branches, each at three different scales. From these results, we can see that each branch is self-similar at the first and second scales because the dimension and regression are almost identical for the largest and medium scales for each branch. Even the atypical branch (shown above) demonstrated self-similarity.
(5)
In addition to the Asparagus plumosus, many plants, such as ferns, trees, and flowers exhibit self-similarity and can be described as fractals. However, plants aren’t the only fractals in nature.
Fractals are found throughout nature and can be used to predict natural phenomena, giving fractals a practical application. While studying the frequency of natural disasters such as earthquakes, landslides, tornados, floods, etc., scientist Kenneth J. Hso deduced a relationship between the frequency and intensity of these natural disasters. As he studied, an inverse log-log relationship between frequency (F) and the intensity of events (M) became apparent. He came up with the following equation: F = c/MD where c is the proportionality constant and D is dimension. Surprisingly enough, the dimension is a fractal dimension and the relationship between frequency and intensity of natural disasters can be thought of as a fractal (6). So if seismic waves can be related to fractals, what about sound waves?
Fractals in Music
One of my favorite toys to play with as a kid was my Talk Girl (a girl’s Talk Boy). With this awesome tape player I could record sounds and then make them slower or faster and laugh at how funny it sounded. With a gadget like a Talk Boy, fractals in music can be identified. Most music isn’t fractal. That is, if it is played at different speeds the music will always sound different. However, some music is fractal. Fractal music sounds the same at every speed, with only adjustments made to the volume of the music. Sounds such as these are called scaling sounds.
Two examples of scaling sounds are white noise and Brownian sounds. White noise, which is the static heard from radios or television sets, is caused by the random motions of electrons through electrical resistance. The noise sounds about the same at all speeds. Brownian noise is caused by Brownian motion, or the random motion of particles in solution. It has been found that this noise is also a scaling sound (8). These two sounds occur naturally, but some musicians actually do compose fractal music.
Fractal music is layered. The same basic principle that applies to creating a drawing of a fractal applies to creating musical fractals. At different levels of the music, self similarity occurs. Mandelbrot described music in the following way:
"The argument that I favor is that musical compositions are, as indicated by their name, composed: First, they subdivide into movements characterized by different overall tempos and/or levels of loudness. The movements subdivide further into the same fashion. And teachers insist that every piece of music be "composed" down into the shortest meaningful subdivisions. The result is bound to be scaling!"(8)
Music has also been analyzed in a more technical manner. Fractal musicsatisfies the relationship F = c/iD, where F is frequency, i is the interval between successive notes, D is dimension and c is a constant. This equation can be rearranged to read
log F = c – D log i. Bach, Invention no. 1 in C Major and Invention no. 13 in A Minor were studied and fractal music was found. Self similar sounds were overlaid to create a complex composition. However, as in most music, only portions of the composition could be described as fractal (6). Most fractal music isn’t the best to listen to. If you want to listen to some, check out the following websites:
To try your hand at composing a piece of your own, check out:
Conclusion
Fractals and fractal geometry are a great new development in mathematics. Fractal geometry makes math more tangible. Before fractal geometry, math was very hard to apply to the real world. As stated in this paper, before fractal dimensioning, objects had to be described with integer dimensions. It was impossible to assign dimensions to real-world objects such as clouds, mountains, flowers, or half-crumpled pieces of paper. However, with fractal geometry this is possible. Fractals can be used in many areas. In addition to nature and music, as seen in the previous pages of this report, fractals are found in statistics, weather, growth patterns, and engineering. They can be used to compress digital images or to create amazing graphics. Fractals help describe things that aren’t completely chaotic, but aren’t completely predictable either.
So when my roommate, Christine, finished reading my paper, she said, “Wow. I never knew so much could come from a single piece of crumpled paper.”
I said, “That’s right: fractals are amazing!”
Works Cited
- Fractal Dimension and Fractals. Vol. 10, No. 4 (2002). Pages 492-434. World Scientific Publishing Company.
- Fractal Geometry of Music. Kenneth J. Hsu; Andreas J. Hsu. Proceedings of the National Academy of Sciences of the United States of America, Vol. 87, No. 3. (Feb., 1990), pp. 938-941.
- Self-Similarity in Asparagus Plumosus. J. R. Castrejon Garcia, Centro de Investigacion en Energa UNAM. March 14, 2002.
- Fractals, Chaos, and Music. Swickape's Math Page.