How many lines will be in the fourth stage of the Cantor dust fractal? The first stage is a single solid line. The second stage has the middle third removed and has two solid lines.(1 point)

Question

Responses

32
32

8
8

4
4

16

Answer 1

To determine the number of lines in the fourth stage of the Cantor dust fractal, let’s analyze the pattern:

1. Stage 1: 1 line (the entire segment).
2. Stage 2: The middle third is removed, resulting in 2 lines (the left and right segments).
3. Stage 3: Each of the 2 remaining segments from Stage 2 undergo the same process. Each segment is divided into 3 parts; the middle part is removed. This results in \(2 \times 2 = 4\) lines.
4. Stage 4: Each of the 4 segments from Stage 3 will have the middle third removed, which results in each segment again being split into 3 parts (the left third, middle third, right third). This means now we will have \(4 \times 2 = 8\) lines.

So, at Stage 4 of the Cantor dust fractal, there will be 8 lines.

The correct response is 8.