Today, used the Box Count method to find again the dimension of Great Britain (report your findings here) then completed one last project to find calculate the dimension of one of the spiral fractal seen on the last dimension calculation sheet (this took most of the remainder of the period).
For Monday, read pages 83-92 in Fractals: The Patterns of Chaos (about fractal math limitations)
If you’d like to rewatch Adam Neely’s Coltrane Fractal video we saw in class (or check out some of his related videos), click the link.
After discussing the reading from the text and the answer to yesterday’s question of the border between Spain and Portugal, we moved on to the last method of finding dimension, the Box Count method.
This method of finding dimension produces the same table of values and log-log plot that we made with the Richardson plot, but the values of S and C are found using a different method. Imagine overlaying a grid on top of a fractal image. We then count (C) the number of boxes of that grid that contain some portion of the fractal. We then repeat this process using a grid with smaller boxes, the sizes of which relative to the original give us S.
The YouTube channel 3Blue1Brown has a great video summarizing all of this.
After enough counts are collected at different scales of boxes, we can create a log(c) vs log(s) plot and find the dimension using the slope as we did before. Your homework is to make the necessary counts with the coastline of Great Britain.
Also homework: For Monday, read pages 83-92 in Fractals: The Patterns of Chaos (about fractal math limitations)
We continued yesterday’s applications of the Richardson Plot to the Koch Curve and finally to the coastline of Great Britain, largely confirming Richardson’s findings as included in Mandelbrot’s article. The results of these can be found here.
Also, at the end of class today, we discussed the border between Spain and Portugal and looked at three maps. Take the data below and answer the following questions:
- What is the dimension of the border between the two countries?
- One country has historically given the length of the border as 987 km, while the other has given a length of 1214 km. Which country is which, and why might this difference have a logical basis (in other words, why might the countries have truly measured the borders in this way? The answer isn’t political!)
Remember: for Friday please read pages 61-73 of your new book Fractals: The Patterns of Chaos
In class today, we derived a new method for finding a dimension of a fractal, the method proposed by Mandelbrot in his article and originally conceived of by mathematician and meteorologist Lewis Fry Richardson. We observed that if we make a “log-log” plot (a so-called “Richardson plot”) of the step sizes and counts of steps that “fit” in a curve, the distribution of points comes out to a roughly linear association, the slope of which is the dimension of the fractal.
We concluded class today by testing this theory to find the dimension of a circle, the results of which can be found here.
We also got a new book, Fractals: The Patterns of Chaos, and our first reading assignment: pages 61-73 (on fractal dimension)