Fractals provide good screen savers, artwork, and lyrics in Disney songs. Some say that a, “fractal is a never-ending pattern” (Fractal Foundation). This means that there is a pattern in place that the shape must follow so that as the shape is enlarged, more of the fractal is revealed. A perfect fractal goes on to infinity and this is what makes it so intriguing. The deeper you look at a fractal the more of it you can see since for a fractal to exist you must be able to look at just part of the fractal and see the entirety of it at the same time. This is an odd concept; however, it makes much more sense while examining a fractal.
The precise definition of a fractal is difficult to determine, because there is not a single definition that is agreed upon. Mandelbrot is revered in the field of fractals and he himself has put out definitions but later recalled them since they were not specific enough for his needs (Feder 2). The most agreed upon part of the definition is that a fractal is a repeated pattern on different scales (Fractals).
Self-similarity, scaling, and dimension are critical to the way that fractals are viewed. Self-similarity is the idea that the closer a fractal is viewed the same shape continues to emerge (Liebovitch 8). This is the pattern repeating itself smaller and smaller as it goes on forever. In order to view these small patterns, scaling is used: relating the size of the shape to the size of the pattern around it which is known to be larger (Liebovitch 8). The scaling factor is the value of the length of the old sections of the fractal in relation to the new pieces. This seems a bit ridiculous: why would we need to know the size of the smaller part of the fractal? The answer is that since all parts of the fractal look identical, there is a need to make sure that it is understood which part of the fractal is being viewed. It also helps to know if someone is referring to the fractal as a whole or a smaller part of the fractal. Dimension takes this scaling information and uses it to determine which parts of the fractals are new by taking the natural logarithm of the number of new pieces of the fractal over the natural logarithm of the scaling factor, (Liebovitch 8). The idea behind dimension is referred to as box counting where a “box” must have greater dimension than the fractal in question, and the dimension is the count of how many boxes it takes to cover the complete fractal (Chapter). This concept can be seen in the picture as a fern that acts as a fractal is covered in boxes and the scales show the smaller parts of the fractal. Self-similarity, scaling, and dimension are how to view fractals so that they can be understood as a whole.
The first piece of fractal music that was written for this project is a basic musical fractal. It starts as six notes: a quarter note followed by four eight notes and another quarter note. At each iteration the longest notes are replaced by a smaller version of the same pattern. As you listen, there are long tones to designate when the next iteration is about to begin. This is a simple fractal meant to demonstrate the concept of a shape that continues to become more and more complex. The piece is a complete fractal because once the most complex iteration is heard, the piece begins to simplify to make the piece symmetric. This confirms that the piece itself is a fractal that grows more complicated in the middle. Please Enjoy.
Played by Eric Lee
One famous fractal is Koch’s snowflake which is traditionally started by an equilateral triangle. However, each individual side behaves the same, so it is easier to discuss the fractal as Koch’s curve. The simplest explanation of the curve is that the line begins as three units long, then the middle unit is replaced by two lines with each of the lines equal to one unit of length (Liebovitch 54). These two lines add a point over the middle third. This pattern continues and with each iteration every straight line gets a point in the middle third. Every iteration is given a number, for example the original line is n=0, after the first point is added it is n=1 and so on every time that the points are added to all of the existing straight lines (Feder 16). The snowflake is the curve just on all three sides of the equilateral triangle expanding on all sides simultaneously. Koch’s curve and snowflake therefore looks like the images below:
(Images from Fractal Foundation and Wikipedia)
From a mathematical point of view, Koch’s snowflake is intriguing. Due to the points being added at every iteration, Koch’s snowflake has infinite perimeter that has a self-similarity dimension (Liebovitch 54). The infinite perimeter comes from the always expanding sides and the self-similarity dimension finds the perimeter’s dimension and we calculate this by dividing the natural logarithm of new pieces (4) by the natural logarithm of scaling factor (3). This comparison of 4/3 also gives the amount that the perimeter grows during each iteration. This means that for each iteration the current perimeter is multiplied by 1.33 to get the new value of the perimeter.
The following piece is a representation of Koch’s Snowflake up to iteration n=3. Between iterations there is three beats of rest so that the listener can understand how much more complicated the piece becomes as n increases. The piece creates the fractal as can be seen visually from the sheet music as the ups and downs of the music match the idea of triangles being added. The longer notes represent the straight lines of the fractal and so in each iteration the long notes were broken up and notes were added in the center to create the feel of a triangular point.
Played by Liv Long
The Sierpinski Triangle is another famous fractal that is studied because of the simplicity of the construction combined with its mathematical aspects. The Sierpinski triangle is created by taking an equilateral triangle and marking points exactly halfway between each of the sides and connecting them (creating an upside down triangle), this idea is repeated to create the fractal (Parsons). The result of the first iteration is 3 smaller triangles within the first one on each side of the upside-down triangle. This continues with three smaller triangles within each of those smaller triangles and the pattern continues. The shape is formed around these upside-down triangles that are “removed” from the shape. This means that there are no additional triangles found in the upside-down triangles. The shape quickly complicates itself and it is easy to see the fractal being produced in the image below. No triangles are added to the spaces that have been removed, but there will always be more space to create new triangles within the remaining triangles.
(Image from Siepinski Triangle by Parsons)
The final fractal piece is a representation of the Sierpinski Triangle. We begin with the bottom line of the triangle: notes are added halfway between the previous notes until there is very little rest in the measure. The rests correspond to the triangles being removed from the larger triangle. The music travels along the bottom of the triangle as described followed by the left side and then the right. The sides are constructed similarly only with rising and falling pitches to match the idea of a triangle. The first measure of each section begins with the end points of the triangles. The result is a building of each side of the triangle separately and then combined for the final line of the song. The result is as follows.
Played by Liv Long