Fractals, Fractal Dimensions and Coastline Paradox investigation

Another example of fractals is the Sierpinski Carpet. Figure 3 is the diagram of the Sierpinski Carpet.

• About the coastline paradox.

The coastline paradox is the counterintuitive observation that states that the length of the coastline does not specify constantly. It also means that the coastline doesn’t have a well-defined definition. This is because coastline has Fractal nature, and has fractal dimensions. Since Fractal dimensions are in between One dimensional and Two dimensional ( as we observed before in the section on Fractals), it is impossible to define the length. This Coastline paradox is proposed by Lewis Fry Richardson, and expanded by Benoit Mandelbrot. The length of the Coastline differs by the method or the equipment it uses since there are too many objects that researchers can use as an index of the smallest scale. Therefore, there are many approximations related to the length of the coastline.

In Mandelbrot’s paper, it says that “Seacoast shapes are examples of highly involved curves such that each of their portions can -in a statistical image of the whole. This property will be referred to as “Statistical Self-Similarity”(mandelbrot). This is the same as fractals

• Relationship between Fractals and the Coastline Paradox

By looking at Fractals, we can say that they have an infinite perimeter with a fixed area. This is similar to the coastline paradox as all coastline has a finite length, thus they have an infinite perimeter, depending on the scale factor “r” that the measuring agency uses. Therefore, I will calculate the fractal dimensions of the two Japanese islands, Hokkaido and Shikoku. I will use two different methods, the “Box counting” method, and the “Hausdorff dimension”. But before I will investigate the coastline of these Japanese Islands, I will discuss the methods to be used.

According to Mandelbrots’s paper, it was calculated that the fractal dimensions of Britain's coastline are about 1.25 ( Mandelbrot’s). Therefore, I will use the number “1.25” as an index to check whether the methods are accurate enough.

Weisstein, Eric W. "Sierpiński Carpet." From Wolfram MathWorld. Web. 20 Nov. 2020.

The percentage error compared to the fractal dimensions that Mandelbrot suggested (1.25) is 66.92% (refer to Equation 5), and this is quite high.

• Equation 5: Percentage Error for the Box counting method.3.77

\(\frac{1.25}{3.77}\times100=66.92\)

I can analyze that this large percentage error is due to the complexity and hardness to plot the boxes around the coastline. It is very hard to manipulate the box. Therefore, the accurate value that we hoped for didn't come out. To add with, I think it is impossible to do this process of plotting the boxes around the coastline (which also takes a lot of time). Therefore, I will not use this method to find the fractal dimensions of Hokkaido and Shikoku. So, I will try another method, which is the Hausdorff dimension method.

• b. Hausdorff dimension

  • i. About the Hausdorff dimensions

The Hausdorff dimension method is introduced by Felix Hausdorff. By using this method, it is possible to calculate the complex coastlines. (Duvall, Keesling, and Vince) This method uses the following formula:

Equation 6: Formula of Fractal Dimension using the Hausdorff dimension

𝐿 = MG       ^1−𝐷

Where:
L = Length of the coastline
G = Measurement Scale
M = Positive Constant
D = Fractal Dimensions

We can explore this equation using the logarithm laws and can convert it to the equation y = ax + b (Study Sinead)

\(log \ L = log\ MG \ \ \ ^{1-D }\)

\(log\ L = log\ M + log\ G\ \ \ \ \ ^{1-D}\)

\(log\ L = log\ M + ( 1 - D)\ log\ G\)

we can expand this equation to the form of y = ax + b

\(log\ L = log\ M + ( 1 - D)\ log\ G\)

\( y\ \ =\ \ b\ \ +\ \ a\ \ \ x\)                   

Therefore​​​​​​​
\(y=log \ L \,\)         \(a=(1 - D)\)         \(\ x= log \ G\ \)         \(b= log \ M\)​​​​​​​

To get the D, which is the fractal dimension, it could be expanded to

- Equation 7 (Calculation to find the Fractal Dimension in Hausdorff dimension)

\( D = 1 -a\)

According to the organization Fractal Foundation, “A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop.”(Fractal Foundation) This means that fractals are a mixture of patterns that never end, and we can make it smaller and smaller infinitely and there’s no minimum size.

There are many examples of fractals, but in this investigation, I will focus on the two most popular fractals; Koch Snowflake and Sierpinski Carpet. To measure the fractal dimensions, the shapes will be scaled by factor and I will be going to count the number of copies of the original versions that are shown.

By looking at figure 3, the middle one is the ⅓ of the original version, and the right one has 8 same geometry as the middle one. Therefore, we can find the shape dimension using this Sierpinski Carpet  We can use the equation to find the fractal dimensions of the Sirenspki Carpet Referring to equation 2.

\(D=\frac {log\ N}{log\ (r)}\\\\D=\frac{log\ 8}{log\ 3}\\D≈ 1.892\)

Therefore, the fractal dimension of the Sierpinski Carpet is about 1.892.

b. Conclusion for Dimensions

In conclusion, by observing the fractal dimensions of the two fractals, Koch Snowflake and Sierpinski Carpet, we find out that they have an infinite perimeter with a finite area. In addition to this by using the equation, \(D=\frac {log\ N}{log(r)}\)we can find out  the fractal dimensions of the fractals. In the next section, I would like to investigate the Coastline Paradox.

Dimension is one of the indexes to show the expanse of space. For example, lines are One dimensional, as there is only one coordinate, pictures are Two dimensional due to x coordinate and y coordinate and our world is Three dimensional as there are 3 coordinates. This is the normal definition of Dimensions, but in this IA, I will focus on the definition of Fractal Dimension by scientist, Mandelbrot. Mandelbrot defined the Fractal dimension as; “If we take an object residing in Euclidean dimension D and reduce its linear size by 1/r in each spatial direction, its measure (length, area, or volume) would increase to N=r^D times the original.” (vanderbilt.Edu). For example, when the D is 2, and r is 2, it creates 4 reduced copies of the original shape. Figure 1 shows how the fractal dimension could be visualized.

• a. Box counting method

  • i. About the box-counting method

According to fractal foundation.org, “The box-counting method is analogous to the perimeter measuring method we used for the coastlines. But in this case, we cover the image with a grid, and then count how many boxes of the grid are covering part of the image”. We can find the fractal dimension using the equation below. (fractal foundation.org) This means that we can find the fractal dimension by changing the scale of the grid that we use to cover the coastline and count the number of the boxes, and input into the equation below.

• Equation 4: Formula of Fractal Dimension using the box counting method

\(D=\frac {log\ N}{log\ r}\)

Where: D = Fractal Dimension 

             N = Number of boxes that cover the pattern

             r = magnification

Dibyendu, Roy. "MATH2916 A Working Seminar." Exploring Fractal Dimension (2019). University of Sydney School of Mathematics and Statistics. Web.

Being a delegate in the Model United Nations(MUN), I often research about the country that I will represent. Two years ago, I was a delegate of Norway. Throughout the research, I realized that the coastline of Norway is longer than Canada. I was very skeptical about this because I knew that the Coastline of Canada should be longer than Norway, by looking at the world atlas. Since I was so curious about this I decided to do additional research about the coastline all around the world. During this research, I realized that depending on the organization that measures the coastline, their methods are so different, therefore the coastline of one country may be different to a great degree, from information from another organization. I wondered, how do people measure the coastline? Are people measuring from the earth? Why is there a great difference in the measurements between the organizations? Isn't it possible to find out the accurate length? To satisfy my curiosity and to find out the answers to my questions, I decided to research this Coastline Paradox. Since my home country, Japan is surrounded by the ocean; I decided to measure the fractal dimensions of two of its islands: Hokkaido and Shikoku.

First, in this investigation, I will study Fractals and Fractal dimensions and the Coastline Paradox and use the concepts to answer my research question, “What are the Fractal Dimensions of the two Japanese islands: Hokkaido and Shikoku?” which I will measure the fractal dimensions of Hokkaido and Shikoku.

Koch Snowflake is one of the most famous types of fractals in the world. Figure 2 is the diagram of the Koch Snowflake