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2. Teachers Notes
This is a brief but very interesting look, at the Von Koch Snowflake Curve. After introducing the curve and discussing its generation, the students are simply asked to derive the perimeter formula for nth iteration.
We then move on to discuss the curve’s finite area and reveal, (by reference to the formula) its infinite perimeter. Students are encouraged to generate a spreadsheet from the formula for the first 50 terms in the sequence to convince themselves of the infinity of the perimeter. (Spreadsheet is at slide 16).
There is a printable worksheet if needed at slide 18 (some students may wish to make jottings/notes on it). If you want to extend still further for the very able, then you might wish to see if they can work out the area An for the nth iteration.

3. The Von Koch Snowflake

4. The Von Koch Snowflake

5. The Von Koch Snowflake

6. The Von Koch Snowflake

7. The Von Koch Snowflake

8. The Von Koch Snowflake P0 = L 1 3 L 1 3 L 1 3 L
The curve is generated from an equilateral triangle by trisecting the sides and constructing this smaller equilateral triangle on each of the sides. This is then repeated ad infinitum. 1 3 L

9. The Von Koch Snowflake P0 = L P1 = 4 3 L 1 3 L 1 3 L 1 3 L
Thinking about the increased length of this side, what will the first new perimeter, P1 be? 1 3 L

10. The Von Koch Snowflake P0 = L P1 = 4 3 L P2 =( )2 4 3 L 1 3 L 1 3 L 1
Derive a general formula for the perimeter of the nth curve in this sequence, Pn. 1 3 L

11. The Von Koch Snowflake P0 = L P1 = 4 3 L P2 =( )2 4 3 L 1 3 1 3
Derive a general formula for the perimeter of the nth curve in this sequence, Pn.
Pn =( )n 4 3 L 1 3

12. P1 = 4 3 L P0 = L A0 A1 A2 A3 P2 =( )2 4 3 L P3 =( )3 4 3 L
The area An of the nth curve is finite. This can be seen by constructing the circumscribed circle about the original triangle as shown. A2 A3
P2 =( )2 4 3 L
P3 =( )3 4 3 L

13. P1 = 4 3 L P0 = L A0 A1 Pn =( )n 4 3 L A2 A3 P2 =( )2 4 3 L P3 =( )3 4
It is a surprising fact therefore that the perimeter of the curve is infinite.
Pn =( )n 4 3 L A2 A3
P2 =( )2 4 3 L
P3 =( )3 4 3 L

14. P1 = 4 3 L P0 = L A0 A1 Pn =( )n 4 3 L A2 A3 P2 =( )2 4 3 L P3 =( )3 4
Whatever fixed value you care to make the perimeter of any curve in the sequence it can always be exceeded by choosing a large enough value for n.
Pn =( )n 4 3 L A2 A3
P2 =( )2 4 3 L
P3 =( )3 4 3 L

15. P1 = 4 3 L P0 = L A0 A1 Pn =( )n 4 3 L A2 A3 P2 =( )2 4 3 L P3 =( )3 4
Use a spreadsheet to compute the first 50 values for the perimeter. Set P0 = 1.
Pn =( )n 4 3 L A2 A3
P2 =( )2 4 3 L
P3 =( )3 4 3 L

16. P0 1 P1
1.333
P26 P2
1.778
P27 P3
2.370
P28 P4
3.160
P29 P5
4.214
P30 P6
5.619
P31 P7
7.492
P32 P8
9.989
P33 P9
13.318
P34
P10
17.758
P35
P11
23.677
P36
P12
31.569
P37
P13
42.092
P38
P14
56.123
P39
P15
74.831
P40
P16
99.775
P41
P17
P42
P18
P43
P19
P44
P20
P45
P21
P46
P22
P47
P23
P48
P24
P49
P25
P50

17. The Von Koch Snowflake
The perimeter of the Von Koch Snowflake Curve is infinite. Just as the coast line of the UK is infinite. The smaller the ruler that you use to measure the coast line, the longer it becomes.
Coast line  miles