The Koch Snowflake
The Koch Snowflake
Math Mock Exploration
Shaishir Divatia
Math SL 1
The Koch Snowflake
The Koch Snowflake is a fractal identified by Helge Von Koch, that looks similar to a snowflake.
Here are the diagrams of the first four stages of the fractal -
1. At any stage (n) the values are denoted by the following –
Nn - number of sides
Ln - length of each side
Pn - length of perimeter
An - Area of snowflake
Mentioned below are the values of these above variables, for the first 4 stages of the fractal.
n
0
1
2
3
Nn
3
12
48
192
Ln
1
Pn
3
4
An
0.57735
0.64150
0.67001
Number of Sides
As the stages of the snowflake progress, each side is divided into thirds, with two equal line protruding from the middle third to form an edge.
I.e – Each straight line -
Becomes this -
Hence now for everyone 1 line, 4 new ones are formed. Hence we can say that there is geometric progression, by the factor 4. Hence, the formula for the number of sides is
Nn = 3(4)n Length of Side
The length of the next side is one-third the previous length. This is once again geometric progression. Therefore, the equation for the nthterm is:
Ln =
Perimeter
The perimeter of any shape = Length of each side x Number of sides
Considering this, the formula for the perimeter can be obtained by multiplying the formulae of the length and number of sides of the fractal.
Hence
Pn = 3(4)n x
Area
The area of every new snowflake would be = area of the earlier stage of snowflake + area of new triangles.
The area of the the first snowflake, or stage zero, is √¾(side)2 . And since the length of the side is 1 unit, the area will be √¾ too.
So to obtain the area for stage two, we add the area of the original triangle, to the area of the three new triangles. The new triangles will have a length scale factor of 1/3, and hence an area scale factor of 1/9.
=0.57735
For the second stage, again the same logic is used. However this time, the length of the new
Math Mock Exploration
Shaishir Divatia
Math SL 1
The Koch Snowflake
The Koch Snowflake is a fractal identified by Helge Von Koch, that looks similar to a snowflake.
Here are the diagrams of the first four stages of the fractal -
1. At any stage (n) the values are denoted by the following –
Nn - number of sides
Ln - length of each side
Pn - length of perimeter
An - Area of snowflake
Mentioned below are the values of these above variables, for the first 4 stages of the fractal.
n
0
1
2
3
Nn
3
12
48
192
Ln
1
Pn
3
4
An
0.57735
0.64150
0.67001
Number of Sides
As the stages of the snowflake progress, each side is divided into thirds, with two equal line protruding from the middle third to form an edge.
I.e – Each straight line -
Becomes this -
Hence now for everyone 1 line, 4 new ones are formed. Hence we can say that there is geometric progression, by the factor 4. Hence, the formula for the number of sides is
Nn = 3(4)n Length of Side
The length of the next side is one-third the previous length. This is once again geometric progression. Therefore, the equation for the nthterm is:
Ln =
Perimeter
The perimeter of any shape = Length of each side x Number of sides
Considering this, the formula for the perimeter can be obtained by multiplying the formulae of the length and number of sides of the fractal.
Hence
Pn = 3(4)n x
Area
The area of every new snowflake would be = area of the earlier stage of snowflake + area of new triangles.
The area of the the first snowflake, or stage zero, is √¾(side)2 . And since the length of the side is 1 unit, the area will be √¾ too.
So to obtain the area for stage two, we add the area of the original triangle, to the area of the three new triangles. The new triangles will have a length scale factor of 1/3, and hence an area scale factor of 1/9.
=0.57735
For the second stage, again the same logic is used. However this time, the length of the new