Difference between revisions of "Fractal Dimension Exploration"

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Here's how to use the Calculator on a Mac to evaluate the expressions in this Exploration:
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Here's how to use the Calculator on a Mac to evaluate the expressions in this Exploration:
  
 
First go to the Applications folder (which you can see from the Dock) and open Calculator.
 
First go to the Applications folder (which you can see from the Dock) and open Calculator.

Latest revision as of 07:16, 28 November 2012


Time-25.svg

Objective: Finding the dimension of fractals.


We can compute the dimension of an object by dividing the sides into r segments (more precisely: the linear scaling factor is r), and counting the number of replacement segments N we obtain:

<math>Dimension = \frac{log N}{log(r)}</math>

(For information on how to use the built-in Calculator on a Mac, see below.)

For example: Take a square, and divide all sides in two (so r = 2). This gives us four smaller squares (so N = 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: r measures the scaling of lines) 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



Here's how to use the Calculator on a Mac to evaluate the expressions in this Exploration:

First go to the Applications folder (which you can see from the Dock) and open Calculator.

Next, under the View drop-down menu, select Scientific (in older Macs, this is Advanced) and Show Paper Tape (i.e., View > Scientific and View > Show Paper Tape).

To calculate, say, log(5)/log(3), hit the keys on the Calculator in this order:

  • 5
  • log
  • divide-key
  • 3
  • log
  • =

The Calculator will show the intermediate results of log(5) and log(3), without showing what's going on; but the Paper Tape will make it all clear.

Then to do another calculation of the same form, just edit the Paper Tape. For instance, to find log(5)/log(2), change the 3 on the Pape Tape to 2, and then hit the Recalculate Totals button on the Paper Tape.


Handin: A sheet with answers to all questions.