Total length of fractals answers:

1. This fractal rule for the fractal Cantor dust is: take out the middle third or each segment in the previous iteration.

A. Draw the next iteration of the fractal

iteration 0
iteration 1
iteration 2
new: iteration 3

B. Complete the table below, and explain your rule for finding the numbers that go in the table

iteration number	number of pieces	length of each piece	total length
1	1	1	1
2	2	1/3	2/3
3	4	1/9	4/9
4	8	1/27	8/27
n	2n	1/3n	2n/3n
...	this doubles each time because a single segment is traded for two smaller segments. after n iterations, you have 2 times itself n times = 2n	the denominator is multiplied by 3 each time OR the length is multiplied by 1/3 each time because when you remove the middle third, the segments left are each 1/3 as long as the previous segment after n iterations, the denominator is 3 times itself n times = 3n	multiply the number of pieces by the length of each piece to get the total length.

 

2. The fractal rule for this fractal (this one doesn't have a name) is: take out the middle third, and replace it with the top 3/4 of a square:

Segment was:	Changed to:

This one's a little hard to visualize, so I have color coded a version to help you see the change from iteration 1 to iteration 2:

Iteration 1 is:	Iteration 2 is:

A. Draw iteration 3

iteration 0
iteration 1
iteration 2
iteration 3

B. Complete the table below, and explain your rule for finding the numbers that go in the table