The Length of the original iteration was given as 1 and with each iteration the fractal divides each side by three. So the side length became a third of the previous. Ln =
The Perimeter of the Koch snowflake was rather simple too. As the length of each side of the snowflake is the same, the perimeter is equal to the amount of sides multiplied by the length of the sides. Pn = Ln x Nn
The area of the Koch snowflake was the most challenging to find. Firstly the area of an equilateral triangle is equal to
. The stage 0 was equal to
as the side length in stage 0 is equal to 1. Every stage after that was equal to the area of the previous iteration, + (Area of an added triangle) x (Number of added Triangles.).
It should be noted that the graph for Pn is divergent. The perimeter has an infinite value when n approaches infinity.
The graph of An is convergent. A has a finite value when n approaches infinity.
Generalizations for behaviour within Graphs.
General Formula for Nn:
The general formula for Nn is 3(4)n. This is because the number of sides is equal to 4 times the previous iterations number of sides.
Verifications:
Nn = N0 x 4n = 3 x 4n
N = 0 3 x 40 = 3 x 1 = 3
N = 1 3 x 41 = 3 x 4 = 12
N = 2 3 x 42 = 3 x 16 = 48
N = 3 3 x 43 = 3 x 64 = 192
General Formula for Ln
The general formula for length is one third of the previous length. The general formula is then 1/3n.
Verifications
N = 0
=
= 1
N = 1
=
N = 2
N = 3
General Formula for Pn
The general formula for perimeter is equal to the Nn multiplied by the Ln. 3(4)n x
. The general formula is then
.
Verifications
N = 0
N = 1
N = 2
N = 3
