The Koch Snowflake

The first 200 words of this essay...

THE KOCH SNOWFLAKE

Q.1) Using an initial side length of 1, create a table that shows the values of Nn, Ln, Pn and An for n = 0, 1, 2 and 3. Use exact values in your results. Explain the relationship between successive terms in the table for each quantity Nn, Ln, Pn and An.

ANS 1)

STAGE

SIDES ( Nn )

LENGTH ( Ln )

PERIMETER ( Pn )

AREA ( An )

STAGE 0

3

1

3

0.433013

STAGE 1

12

0.3333333333

4

0.577350

STAGE 2

48

0.1111111111

5.3333333333

0.641500

STAGE 3

192

0.037037037

7.1111111111

0.670011

At first, let us take a look at the change in the sides of the snowflake with increase in stages. It is soon realized that with increase in stage, the snowflake structure changes and so do the sides consequently. On further observation, it is seen that the snowflake sides increase in geometric progression.

This is seen because:

12/3=48/12=192/48=4:1

4:1 is the constant ratio. The sides increase by four times with