Difference between revisions of "Fractal Dimension Exploration"

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{{Exploration}}
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{{Time|25}}
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{{Objective|Finding the dimension of fractals.}}
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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:
 
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:
 
  <center><math>Dimension = \frac{log N}{log R}</math></center>
 
  <center><math>Dimension = \frac{log N}{log R}</math></center>
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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.
 
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.
  
[[Image:Sierpinski4.png]]
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[[Image:Sierpinski4.png|150px]]
 
   
 
   
 
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.)
 
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.)
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3. There is a variation on the snowflake called the Zig-Zag.
 
3. There is a variation on the snowflake called the Zig-Zag.
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We divide a line segment into four congruent parts. Replace the two segments next to the middle by two new segments as indicated.
  
 
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[[Image:Zigzag.png]]
 
 
  
  

Revision as of 13:19, 24 July 2007


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. Find the dimension of 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. What is the dimension of 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. There is a variation on the snowflake called the Zig-Zag. 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