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> | ||
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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). | ||
| + | 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]] | |
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: | + | 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. | ||
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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 | + | c. Calculate the dimension. |
| − | + | ||
| + | 2. What is the dimension of the Koch edge? | ||
| + | [[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:
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. Find the dimension of the Sierpinski Triangle:
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?
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
