In forming the Koch snowflake in Figure 8.104, the perimeter becomes greater at each step in the process. If each side of the original triangle is 1 unit, a general formula for the perimeter, L, of the snowflake at any step, n, may be found by the formula.

Figure 8.104:

L = 3(4/3)^{n - 1}

For example, at the first step when n = 1, the perimeter is 3 units, which can be verified by the formula as follows:

L = 3(4/3)^{1 - 1} = 3(4/3)⁰ = 3 •1 = 3

At the second step, when n = 2, we find the perimeter as follows:

L = 3(4/3)^{2 - 1} = 3(4/3) = 4

Thus, at the second step the perimeter of the snowflake is 4 units.

(a) Use the formula to complete the following table.

Step	Perimeter
1
2
3
4
5
6

(b) Use the results of your calculations to explain why the perimeter of the Koch snowflake is infinite. 

(c) Explain how the Koch snowflake can have an infinite perimeter, but a finite area.

Transcribed Image Text:

Step 1 Step 2 Step 3 Step 4