Tag Archives: Coastline Paradox

Fractals & Chaos Recap for 10/22

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:

  1. What is the dimension of the border between the two countries?
  2. 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!)
Step Size S C Distance measured
100 km 1 7.3 730 km
50 km 2 16.2 810 km
25 km 4 35.4 885 km
10 km 10 93.2 932 km
5 km 20 200.6 1003 km

Remember: for Friday please read pages 61-73 of your new book Fractals: The Patterns of Chaos

Fractals & Chaos Recap for 10/18

If you have a compass (the circle-drawing kind) please bring it with you to class over the next few days.

We discussed Mandelbrot’s Article, then used it to segue into a further discussion of dimension. As we are aware, we still have problems with the Hausdorff dimension formula for calculating dimension of fractals. It can’t handle fractals with stems (i.e., non-iterating segments that never disappear) and with fractals that are not exactly self-symmetric.

Today, we considered a football field, a circle, and a Koch Curve, and looked at how the size of the measuring stick we use to measure the length or perimeter of such things has an impact on the total amount of length we actually calculate. For a football field, the size of the stick makes no difference. We’ll be obtaining 100 yards worth of length even if we use a foot (S = 3) or an inch (S = 36) as our step size.

For a circle, this isn’t the case. Use a measuring stick the length of the diameter, and we can only make two steps before we end where we started. Use a stick the size of the radius (S = 2) and we can make 6 such steps (resulting in a measure of three diameters). Use a half-radius (S = 4), and we wind up with a total length of slightly more than 3 diameters. There is a limit to this, of course: pi*d, which is precisely the formula for the circumference of a circle.

For the Koch Curve, the story is very different. Use a step size the length of the original baseline, and we can make one step. Use a step size of 1/3 the baseline (S = 3), we can make 4 steps, giving a length of 4/3 the base. Use a step size of 1/9 the baseline (S = 9), and we can make 16 steps, for a total length of 16/9 the base. As we shrink the length of the ruler we use, the number of steps increases more quickly, and so the total length increases without bound.

We’ve seen suggestions at this idea before. In the second article we read (The Diversity of Life), we saw that reducing the scale of our perspective dramatically increases the amount of living space we can find. This idea is also found at the center of the coastline paradox, hinted at in the Ants in Labyrinths article (see also this blog post from UK Urban Planner Alasdair Rae)

We will be expanding on this in class tomorrow, including a discussion on what all this has to do with the dimension of what we’re measuring.