The Dimension of Fractals

The important video that you should watch before reading this article: https://www.youtube.com/watch?v=gB9n2gHsHN4&ab_channel=3Blue1Brown

In this article, I will write about how I calculated the dimension of the coastline of Taiwan.

I put an image of the Taiwan map with a scale factor of 1 into GeoGebra, and after that, I put 1*1 squares on the places the coastline had touched. As a result, there were 27 squares that the coastline had touched. In the end, it looked like Fig 1, and the number of squares that had touched the coastline in different scale factors is in Table 1.

Fig 1

Table 1

According to the video I posted, the graph will be the log of the scaling factor on the x-axis and the log of the number of squares that had touched the coastline on the y-axis, like in Fig 3. So the result should be like in Table 2.

Fig 2

Table 2

After drawing the prediction line of Fig Table out, I noticed something was wrong. According to the video I provided, the slope of the prediction line is the dimension of the picture, but the slope of the prediction line is about 0.89, while I think it will be over 1. Maybe I had counted wrong when the number was too big, or there wasn't enough data. Another possible reason is that the squares are not very precise. I should make them smaller.

Fig 3