Finding Infinity in a Snowflake

Cynthia S. Thomas, David A. Thomas, Michelle Wallace

Assumptions

This activity assumes that the user has access to and an operating knowledge of the following technologies:

· WWW Browser	Microsoft Internet Explorer 5.0 +
· Spreadsheet Program	Microsoft Excel 2000 +
· Geometers Sketchpad	Version 3.0 +

Part I Background Information

1. Read about Niels Fabian Helge von Koch at the MacTutor History of Mathematics Archive.

2. Read about and experiment with the Koch Snowflake Applet at the Shodor Education Foundation.

1. What happens to the shape of the figure from one level to the next?
2. What happens to the number of segments from one level to the next?
3. What happens to the length of each segment from one level to the next?

The Koch Snowflake

Part II Investigating curve length using the Geometers Sketchpad

3. Start the Geometers Sketchpad. Open the file flake1a and drag its window to the right side of the screen.

4. Open a new sketch. Construct and select two points in the new sketch.

5. Click the Fast button in the flake1a window. When the Recursion window opens, enter 0 and click OK

6. You should get an image like that shown below. Measure the distance between the two endpoints and adjust that distance to 9.000 inches.

7. Note that this portion of the snowflake curve consists of four segments of equal length. Select one of the segments and measure its length. Record the number of segments and the segment length in the Excel spreadsheet Segment Length and Perimeter model. The spreadsheet will automatically compute the total curve length for recursion level 0.

Depth of Recursion	Segment Length	Change factor (segment length)	Number of Segments	Change factor (number of segments)	Total Curve Length	Change factor (Total Curve Length)
0
1		From 0 to 1		From 0 to 1		From 0 to 1
2		From 1 to 2		From 1 to 2		From 1 to 2
3		From 2 to 3		From 2 to 3		From 2 to 3
4		From 3 to 4		From 3 to 4		From 3 to 4

8. Select your original two points and repeat the process using the next recursion level.

9. Continue gathering data using subsequent recursion levels.

10. As the depth of recursion increases, it becomes more and more difficult to select, measure, and count segments. What mathematical patterns do you see in the data that could be used to extend the table in the spreadsheet to model the snowflake’s features at higher recursion levels? Specifically, what do you think the change factor should be from any level to the next for …

a. Segment length? Why?

b. Number of segments? Why?

c. Total curve length? Why?

11. Imagine the path that you would obtain if you could enter 1000 as the recursion level. If you entered 1,000,000. Could you trace these curves? Why or why not?

Part III Investigating area inside the curve using the Geometers Sketchpad

12. Open flake1 and repeat steps 3, 4, and 5 in Part II. You should get a figure like that shown below:

13. Highlight the three vertices of the largest triangle, construct the polygon interior and measure its area. The figure should look that that shown below.

14. Record the measurement in the MS Excel spreadsheet model Area Inside the Curve.

15. On subsequent recursion levels, you will construct, measure, and count smaller and smaller triangles. For instance, after constructing a triangle at recursion level 1, your Geometers Sketchpad model should include shaded triangles from both level 0 and level 1 (See below).

16. Take measurements at as many levels of recursion as possible. Look for patterns in your data that would enable you to extend the model to higher levels of recursion.

Depth of Recursion	Area of one new triangle	Change Factor (Area of one new triangle)	Number of new triangles	Change Factor (Number of new triangles)	Previous Area	Newly Added Area	Change Factor (Area)	Total Area
0
1		From 0 to 1		From 0 to 1			From 0 to 1
2		From 1 to 2		From 1 to 2			From 1 to 2
3		From 2 to 3		From 2 to 3			From 2 to 3
4		From 3 to 4		From 3 to 4			From 3 to 4

17. As the depth of recursion increases, it becomes more and more difficult to select, measure, and count triangles. What mathematical patterns do you see in the data that could be used to extend the table in the spreadsheet to model the snowflake’s features at higher recursion levels? Specifically, what do you think the change factor should be from any level to the next for …

a. Area of one new triangle? Why?

b. Number of new triangles? Why?

c. Newly added area? Why?

d. Total area? Why?

18. Imagine the path that you would obtain if you could enter 1000 as the recursion level. If you entered 1,000,000. Could you color the interior of these curves? Why or why not?