Question

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 base. Step 3: Repeat Step 2 infinitely many times. We will stop after 3 iterations- this should be enough repetitions to give you an idea of the final outcome. However, feel free to do more iterations - if you can! a) Complete the following table: Assume the original triangle has side length 1 unit, and the area is A square units. Step (n) Number ofArea of new triangles snowflake (An) Length of one side (Sn) Number of Perimeter (p) Area of one new triangle (an) sides 3 3 0 12 2 3 5 Using series notation, answer parts (b) and (c): h) What happens to pn, the perimeter, as n goes to infinity? c) What happens to An, the total area of the snowflake, as n goes to infinity?

Answer 1

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

$P_n = 3(\frac{4}{3})^n$

So, when n tends to infinity, hence the perimeter will tend to infinity

c) The area of the Koch flake will be

$A_n = A_{n-1} + \frac{1}{3}(\frac{4}{9})^{n-1}A$

where A is the area of the initial triangle

Solving this recurrence relation we get

9 3 3

So, when n tends to infinity, the area will be tending to 8/5 * A

Note - Post any doubts/queries in comments section.