Fractal Dimension Exploration

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Time-25.svg

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:

<math>Dimension = \frac{log N}{log R}</math>


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.

Sierpinski4.png

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:

Sierpinski-steps.svg

a. What is r (scaling ratio)?

b. What is N (number of pieces we keep)?

c. Calculate the dimension.


2. The Koch edge?

Koch1.png

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.

Zigzag.png


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 the dimension



Handin: A sheet with answers to all questions.