If you have a compass (the circle-drawing kind) please bring it with you to class over the next few days.
We discussed the assigned reading from Fractals: The Patterns of Chaos, then segued 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.
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.