Example 3

The equilateral triangle below is Stage 0 of a Koch Snowflake. Draw Stage 1 of the snowflake by first drawing an equilateral triangle on the middle third of each side. Then erase the middle third of each side of the original triangle.

• To draw an equilateral triangle on the middle third of a side, find the two points that are 1 3 1 third  unit from an endpoint of the side. From each point, draw a segment of length 1 3 1 third  unit. Each segment must make a 60 ° 60 degrees  angle with the side of the original triangle.

  

  

Exercises

1. Draw Stage 3 of the fractal tree in Example 1.

Use the Koch Curve in Example 2 for Exercises 2–4.

2. Complete the table to find the length of the Koch Curve at each stage.

   Stage	0	1	2
   Length	1	□ white square	□ white square

3. Examine the results of Exercise 2 and look for a pattern. Use this pattern to predict the length of the Koch Curve at Stage 3 and at Stage 4.
4. Suppose you are able to complete a Koch Curve to Stage n.
   1. Write an expression for the length of the curve.
   2. What happens to the length of the curve as n increases?
5. Draw Stage 2 of the Koch Snowflake in Example 3.

Stage 3 of the Koch Snowflake is shown below. Use it and the earlier stages to answer Exercises 6–8.

6. At each stage, is the snowflake equilateral?
7. 1. Complete the table to find the perimeter at each stage.

      Stage	Number of Sides	Length of a Side	Perimeter
      0	3	1	3
      1	□ white square	1 3 1 third	□ white square
      2	48	□ white square	□ white square
      3	□ white square	□ white square	□ white square

   2. Predict the perimeter at Stage 4.
   3. Will there be a stage at which the perimeter is greater than 100 units? Explain.
8. Exercises 4 and 7 suggest that there is no bound on the perimeter of the Koch Snowflake. Is this true about the area of the Koch Snowflake? Explain.

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