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The Koch snowflake                                       

Introduction: The Koch snowflake fractal is built by starting with an equilateral triangle, and removing the inner third of each side, building another equilateral triangle at the location where the side was removed.  When n = 0, 1, 2and 3 each value of Nn, ln, Pn and An can be shown as the following table:

And the process I use to get the value are shown below:

The number of the side:

To get the next snowflake, we can found that, each side of the triangle will break into 4 new sides, hence the number of each side will always 4 times than previous one. Thus we get

                                                                Nn=4Nn-1

 Thus we assume that Nn is the single of the number of sides and get the process below:

        

                     Shape1: we get a triangle. N1=3

                                            Shape2: We break into 4 new sides. So the N2=3×4        

                                            Shape 3: Each side then brake into 4 new sides, So N3=3×4 ×4

                                           

                                             Shape 4:We repeat the same step, thus we get N4=3×4×4×4

                                         

And then we graph our data:

                                                     The number of the side

We can found that the shape of the graph is a geometric series. Thus, if we want to get the next shape, we should time the same number. Thus it’s a geometric series which has a first term of 3 and a common ratio of 4, obviously. So we can get the geometric formula:

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Then, to verify the formula, we place

n=0                     n=1                    n=2                    n=3    

And then we get:

N0==3     N1==12   N2==48   N3==192

The results are already proved above.

The length of each side:

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