Besides being beautiful, fractals are important and can be used in the world all around us.
Some of those areas include: "applications of the fractal equation in studies of climatology, coastlines, soil erosion,
mountains, seismic patterns and other aspects of nature such as: snowflakes, ferns, the branches of a tree, and the florets of a cauliflower" (Padula, 2005).
Although all are important to look into, I want to focus on the use of fractals in music and the measurement of coastlines.
Fractals and Music
In general mathematics is a least favorite subject among students. It can make it hard for teachers to teach the subject since students
have lost interest and have a hard time relating it to applications they enjoy. Well using fractals and techno music is one way to spark an interest in students.
Padula mentioned that "they (students) can also learn about the importance of pattern to mathematics by listening for patterns in mathematically structured music, such
as fractal music"(2009). Techno music is a great example of fractal music that is in the world around us. To start dissecting how the music works it is important to
understand that even though fractals are graphed in order to see their recursive properties, we look at a different parameter in order to "hear" the patterns in music.
"The parameters in fractal music are sonic rather than visual and can include: pitch, rhythmic values and dynamics (in music, variations in the volume of sound)" (Padula, 2005).
This means that instead of viewing the recursive properties of fractals, we turn them into properties of music that technology like computer synthesizers use to create
techno music. One way to think about it is the physics of a sound wave. If you were to place a needle on a tuning fork and ring it, the needle would quiver in a
sine function pattern. This is how a sound wave works. The different amplitudes and periods of different tuning forks together create the sounds that we hear, but they
can be described by these mathematical patterns we are familiar with. In order to make it fractal music though, you must use your output sound in order to start your new input sound.
You will then be able to map what you have created to notes. If you have created fractal music you will be able to see a self-similar pattern in the write up of the music.
Here is a picture indicating an example of what that might look like (Padula, 2005):
Even without being able to read music you can see that the three different notes patterns all follow the same steps. As you take notes away, or add more, you will get the
same melody that you started with, although the more notes you add the smoother it might sound. Other than looking at the music sheet, we can also look at the use of
the recursive formula that can produce a table of repeating values that a computer synthesizer can turn back into sound. The following chart is created by the following
pattern described by Padula in her 2009 article: "Take any whole number A, a constant multiplier B and a number which will
be the maximum number of digits allowed C. Next, multiply A and B. Take the result and multiply it again by B. Repeat this process until the number of
digits in the result get past C. Now sever the result by truncating it to its rightmost C digits. Multiply again by B and sever again to C digits, and so
forth. It turns out that all earthworms (mathematically speaking) eventually enter a cycle, an infinite loop of repeating values" (Padula, 2009)
A visual representation would look like so:
This chart alone provides the opportunity to create many different musical patterns that are all fractal, and you can see at the end of the chart how this formula gives
us a recursive formula, and thus fractal music. Overall, it is important to keep in mind that for fractal music we are looking for music with a recursive sound. One that
is repeated over and over and if you were to look deeper into the musical notes you should be able to identify patterns similar to the parent pattern of musical notes,
but with more detail.
Fractals and Measuring Coastlines and Borders
Measuring coastlines has been a long application of fractals. For instance, there is the problem about measuring the coast of Great Britain.
Well there was a man by the name of Lewis Fry Richardson who faced a similar problem. McCartney, Myers & Sun stated his problem in their article:
"the Spaniards reckoned their border with Portugal was 987 km, whereas the Portuguese reckoned the same border to be 1214 km. The issue at stake was that if Portugal and Spain could not agree on the
length of the border – perhaps it was because they were not agreeing on where the border was and hence whose land was whose. Richardson resolved the issue [8]
by concluding that each side had simply used measuring sticks of different length(rather than the unsavory alternative of one nation or the other sneakily moving the
border around)" (2008). As Richardson could see, if you were to measure the border of Portugal with different measuring devices you would get different lengths.
This is an application of fractals seen by cartographers. The more detailed one draws the coastline or border of a country, the more length there would be in the length
of the line drawn, and in theory there might not be any length at all. An interesting application that Richardson used was this knowledge of measuring borders or
coastlines and the dates of maps. He started looking at maps that were not modern (meaning they were dated before 17th century Ireland) and noticed that if you
measured them they appeared to have less length the older they were since there was less detail in the drawings. He was able to create a coefficient to describe this
relationship called a divider dimension. Much like a correlation coefficient in statistics, it mapped the length of a border or coastline to the age of the map.
He was able to discover that there exists a relationship between map age and length of the border, so that one might be able to predict the age of a map depending
on how the length related to more recent maps. Branching off of this idea, there became talk of coastlines not even having length. "...the [divider dimension] characterizes the complexity of the coast of Britain
over some range of scales by expressing how quickly length increases if we measure even finer accuracy. Eventually, such measurements do not make
much sense anymore because we would run out of maps and would have to begin measuring the coast in reality and face all the problems of identifying
where a coast begins and ends, when to measure (at low or high tide), how to deal with river deltas and so on. In other words the problem becomes
ridiculous. But nevertheless, we can say that in any practical terms the coast of Britain has no length. The only meaningful thing we can say about its length is
that it behaves as a power law over a range of scales to be specified and that this behaviour will be characteristic" (McCartney, Myers, Sun, 2008).
Although one has right to agree or disagree, I find it interesting to consider that if we continually look at smaller measuring devices we could discover that many different
lengths and borders are actually all the same at some degree. You could use this idea to fuel a class discussion with some authentic learning about the application
of fractals. If you look at the Resources link you will find a copy of a lesson plan and worksheet to give students in order to explore this idea.
