1. The Cantor Set: an In-depth Gander at Fractals
Greg Blachut
Keith Janson
Kinan Hayani
Jake Folkerts

2. History The Cantor Set was first published in 1883.
It is named after the German Mathematician Georg Cantor.
It is probably the most important early mathematical set.

3. An example of “Cantor’s Dust”
You take a line (no depth) of x length.
Divide it into 3 pieces, and take out the middle one.
Repeat

4. Formula for number of segments
How many segments are there at step 1?
How many segments are there at step 2?
How many segments are there at step 3?
How many segments are there at step 4?
So you have 1, 2, 4, 8, . . .

5. So the formula for number of segments is ? ? ?
“Two to the n minus one”

6. The length of a segment
Okay, so remember that “you divide it into three parts and take out the middle part”.
So, lets say the first part’s length is X.
So after dividing it into three parts, and taking out one segment, what is the length?
The total length at step two is 2/3x.
Remembering the last formula, how many segments are there at step two?
So how long is each individual segment?

7. Now the equation . . .
So, the first segment is X long, then in the next iteration, each segment is 1/3x, and then the next one is 1/9x.
So again, the lengths is a geometric sequence.
The Length of an Individual Segment:
(x is the original length)
What happens to the sequence as “n” goes to infinity? What does that mean?

8. To Find the Total Length of the Segments . . .
The total length = (the length of one) * (the total amount of segments)
What happens to the sequence as “n” goes to infinity? What does that mean (again)?