Fractal Dimension Exploration
Objective: Finding the dimension of fractals.
We can compute the dimension of an object by dividing the sides into r segments, and counting the number of replacement segments N we obtain:
For example: Take a square, and divide all sides in two (so r is 2). This gives us four smaller squares (so N is 4).
Then we obtain: <math>Dimension = \frac{log 4}{log 2} =2</math> . This is no great surprise of course, but this (strange) formula will help us compute the dimensions of self similar fractals.
Suppose we take a triangle and divide all sides in fourths (r = 4) and we keep 10 of the small triangles (N = 10). (There are 16 small triangles, but we keep only 10 of them.) We obtain: <math>Dimension = \frac{log 10}{log 4} = 1.66</math>
This means that if we repeat the process (i.e. divide each small triangle into 16 smaller ones and keep only 10 of those), then the resulting shape has dimension 1.66.
A fractals is an objects whose dimension is not a whole number, hence the name fractal.
1. The Sierpinski Triangle:
a. What is r (scaling ratio)? b. What is N (number of pieces we keep)? c. Calculate the dimension.
2. The Koch edge?
a. What is r (scaling ratio)? b. What is N (number of pieces we keep)? c. Calculate the dimension.
3. The Zig-Zag curve.
We divide a line segment into four congruent parts. Replace the two segments next to the middle by two new segments as indicated.
a. Draw one more iteration of the Zig-Zag curve.
b. What is r (scaling ratio)?
c. What is N (number of pieces we keep)?
d. Calculate
