Difference between revisions of "Fractal Dimension Exploration"

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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>
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).
 
 
  
  
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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).
  
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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: . This is no great surprise of course, but this (strange) formula will help us compute the dimensions of self similar fractals.
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[[Image: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.)
 
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:   
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We obtain:  <math>Dimension = \frac{log 10}{log 4} = 1.66</math>
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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.
 
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.
Definition: A fractals is an objects whose dimension is not a whole number, hence the name fractal.
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A fractals is an objects whose dimension is not a whole number, hence the name fractal.
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 +
 
 
1. Find the dimension of the Sierpinski Triangle:
 
1. Find the dimension of the Sierpinski Triangle:
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 +
[[Image:Sierpinski-steps.svg]]
 
   
 
   
 
a. What is r (scaling ratio)?
 
a. What is r (scaling ratio)?
 
b. What is N (number of pieces we keep)?
 
b. What is N (number of pieces we keep)?
c. Calculate  
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c. Calculate the dimension.
2. What is the dimension of the Koch snowflake?
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2. What is the dimension of the Koch edge?
  
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[[Image:Koch1.png]]
  
 
a. What is r (scaling ratio)?
 
a. What is r (scaling ratio)?
 
b. What is N (number of pieces we keep)?
 
b. What is N (number of pieces we keep)?
c. Calculate  
+
c. Calculate the dimension.
 +
 
 +
 
 
3. There is a variation on the snowflake called the Zig-Zag.
 
3. There is a variation on the snowflake called the Zig-Zag.
  

Revision as of 13:15, 24 July 2007

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.



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