The Koch Snowflake

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        

3. The graph for Number of sides supports the generalisation that to calculate the proceding value of number of sides, the preceding value must be multiplied by 4.

It can be deduced from the graph of Side length that 1/3n can be applied to all values of n to calculate the length of the sides of the fractal. Verifying it with the values of side length in the table :

        

                        

n = 0 : 1/30 = 1

                        n = 1 : 1/31 = 1/3

                        n = 2 : 1/3² = 1/9

                        n = 3 : 1/3³ = 1/27

The perimeter is found using the formula Nn x ln and it is clear from the graph that as the values of n increase, the perimeter increases almost constantly.

The area increases with the values of n but the difference between each successive value decreases.                                                                                 

4. N4 = preceding number of sides x 4 =192 x 4 = 768

L4 = 1/3n = 1/34 = 1/81

P4 = Nn x ln = 768 x 1/81 = 12.64

A4 =3 .4.4.4.√3/4 . (1/34)² = 0.6876

5. Values for n= 6 :-

        

        N6 = 3072 x 4 = 12288

        L6 = 1/36 = 1/729

        P6 = 12288 x 1/729 = 16.866

        A6 =  3.4.4.4.4.4 .√3/4 . (1/36)² = 0.0025 + 0.6876 = 0.6901

6.

                

        A pattern can be clearly seen that the values continue increasing however as the n value gets larger, the difference between successive values decreases.

7. When the perimeter of a geometrical shape increases, the area normally increases aswell, however in this case it occurs differently.

As the value of n gets larger and larger, the perimeter keeps growing infinitely, e.g :-

        

The area however, keeps growing approaching a number but never exceeding it : -

Therefore as n -> ∞, the difference between the values of area decreases more and more.

Hence a general formula for area can be found by using the same equation used to find the areas, however adding all of the values up:

√3/4 x l²(1 + ∑nn=1 3x4n-1/9n )

The area is limited while the perimeter is infinite.