Von Koch's Curve

Mathematics of the Von Koch Snowflake

1. Suppose C1 has a perimeter of 3 units. Find the perimeter of C2, C3, C4, and C5.Remember that Von Koch's  curve is Cn, where n is infinitely large, find the perimeter of Von Koch's Curve.

2. Suppose that the area of C1 1 unit². Explain why the areas of C2, C3, C4, and C5 are  
A2=1 + 1/3 unit²                            
A3=1 + 1/3 (1 + 4/9) unit²  
A4=1 + 1/3 (1 + 4/9 + (4/9) ²) unit²  
A5=1 + 1/3 (1 + 4/9 + (4/9) ² + (4/9)³) unit²     
A6=1 + 1/3 (1 + 4/9 + (4/9) ² + (4/9)³ + (4/9)⁴) unit²     
A7=1 + 1/3 (1 + 4/9 + (4/9) ² + (4/9)³ + (4/9)⁴ + (4/9)⁵) unit²     

Use your calculator to find An where n=1,2,3,4,5,6,7,etc.,giving answers which are as accurate as your calculator permits. What do you think will be the area within Von Koch’s snowflake curve?

1.There are many ways you can find the perimeter of the von koch snowflake. You can use a formula, the way we showed you in the video below, etc.
In our project for the IB math SL we used both the long complicated way showed in the video and the formula.
Remember, the formula for the square curve is different from the snowflake formula.
Cn=4,5 1/3,7 1/9, 9 13/27, 12 52/81

2.To find the area the only thing we used was the way we showed in the video. If the do it the way we showed you in the video and want to have the same answers as the book, once you find the are of a small triangle when you add it do not collect the like terms use backwards foil to get an answer like the ones to the left.