HandsOn 8 - Creating Your Own Mathematical Fractal

The mathematical fractal coastline we are about to create is called the "Koch curve.'' You will need:

In this procedure, we follow the same rule over and over. The central third of a rubber band segment gets turned into a V, like this:


Figure 2.9: Beginning the construction of the mathematical fractal called the Koch curve.

Table 2.7: Properties of a Koch Curve.
Length of each step Number of steps to follow coastline Total length
(cm.) exactly (cm.)
9 1 9
3 4  

Q2.13: Imagine carrying out the process of turning the middle third of each segment into a V again and again an infinite number of times. How would the curve look? How long would each step be? How many steps would it take to follow the coastline exactly? And what would be the total length of the coastline?

Q2.14: How does this mathematical fractal differ in appearance from the random fractal coastlines you created in the exercises above?

Plot the table entries on a copy of the log-log graph paper of Figure 2.2. (Note: Recall that you can multiply all entries on both axes by 10 or 1/10 to accommodate the range of numbers to be plotted.) The length of each segment is plotted along the horizontal axis and the number of steps along the vertical axis. Is the result a straight line? If so, what is the slope of this line? The accepted value of the dimension of a Koch curve is 1.26. How does the magnitude of your measured slope compare with this number?

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