a) The table data will be
Step (n) | Length of one side (Sn) | Number of sides | Perimeter (Pn) | Area of one new triangle | Number of new triangles | Area of snowflake (An) |
0 | 1 | 3 | 3 | n/a | 0 | A |
1 | 1/3 | 12 | 4 | 1/9A | 3 | A + 1/3A |
2 | 1/9 | 48 | 16/3 | 1/81A | 12 | A + 1/3A + 4/27A |
3 | 1/27 | 192 | 64/9 | 1/729A | 48 | A + 1/3A + 4/27A + 16/243 A |
4 | 1/81 | 768 | 256/27 | 1/6561A | 192 | A + 1/3A + 4/27A + 16/243A + 64/2187A |
5 | 1/243 | 3072 | 1024/81 | 1/59049 | 768 | A + 1/3A + 4/27A + 16/243A + 64/2187A + 256/19683A |
n | 1/3^n | 3(4)^n | 3(4/3)^n | 1/2^(2n)A | 3(4)^(n-1) | A(n-1) + 1/3*(4/9)^n-1 * A |
b) The perimeter of Koch flake is given by the formula
So, when n tends to infinity, hence the perimeter will tend to infinity
c) The area of the Koch flake will be
where A is the area of the initial triangle
Solving this recurrence relation we get
So, when n tends to infinity, the area will be tending to 8/5 * A
Note - Post any doubts/queries in comments section.
Part II - Snowflake Island 0r The fractal called snowflake island (or Koch's snowflake) is constructed as follo...
Part II - Snowflake Island 0r The fractal called snowflake island (or Koch's snowflake) is constructed as follows be You will make foam versions of each iteration you create. It will help iteration as a pattern for you to cut out of the foam. Step 1: Begin with cutting out an equilateral triangle. to make paper versions of each rever you see a straight line, draw an equilateral triangle on the middle third of the line segment and erase its...