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Cantor's Comb Exploration Questions
The Cantor comb is a way to visualize the famous Cantor set.
Georg Cantor was interested in this set because it is infinite
and completely disconnected (all of the points are separated
from each other), even though it is built by looking
at line segments. In fractal terms we refer to such disconnected
sets as "fractal dust."
Directions: Draw several iterations of the basic Cantor Comb (removing the middle 1/3 each
time), filling in the table below.
Cantor's Middle Thirds Comb
| Iteration |
1 |
2 |
3 |
4 |
5 |
| Length of one line segment |
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| Total Length |
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Answer the following questions:
- What would the length of a line segment in the N-th iteration be?
Look at the patterns made by the numbers
both before and after simplifying.
- What would the total length be in the N-th iteration?
- What do you expect the Cantor Comb to look
like? In other words, what would you expect to happen if you
repeated this infinitely many times?
- What is the length of the Cantor Comb?
Now try other fractions:
Try removing the middle 1/2 and the middle 1/4, filling in the
tables below:
Cantor's Middle Halves Comb
| Iteration |
1 |
2 |
3 |
4 |
5 |
| Length of one line segment |
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| Total Length |
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Cantor's Middle Quarters Comb
| Iteration |
1 |
2 |
3 |
4 |
5 |
| Length of one line segment |
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| Total Length |
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Compare the results of these trials. Try a few other fractions
of your choice and then list any general conclusions you can
draw. For example, can you build a formula for the N-th iteration
total length that would work for any fraction of the form
1/X?
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