HL type 1 portfolio on the koch snowflake

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                                                                                       HL TYPE 1

                                 

                                                                                   By Abhishek Chhabria

                                                                                    (Math HL)  

                                                                         International Baccalaureate

                                                                       B.D.Somani International School

INTRODUCTION

The Koch curve was named after Helge von Koch in 1904. The generation of this fractal is simple. We begin with a straight line of unit length and divide it into three equally sized parts. The middle section is replaced with and equilateral triangle and its base is removed. After one iterations, the length is increased by four-thirds. As this process is repeated, the length of the figure tends to infinity as the length of the side of each new triangle goes to zero. Assuming this could be iterated an infinite number of times, the result would be a figure which is infinitely wiggly, having no straight lines whatsoever. The construction is very simple but still looks really beautiful.

Throughout, let us consider-

  1. Using an initial side length equal to 1, we deduce a table which shows the values of the above mentioned variables with respect to changes in ‘n’.

     Stage 0                   Stage 1                  Stage 2                  Stage 3

Now we study the relationship between successive terms for each of the above geometric deductions of Koch’s snowflake.

Let us consider the term in the stage to be respectively, according to the general term being studied, and ‘r’ to be the ratio between successive terms.

  • For ,

To confirm the value of ‘r’ we use a new pair of terms,

Therefore, we conclude that the ratio between the successive terms of the geometric progression is 4.

  • For ,

        

Therefore we come to the conclusion that the ratio between the successive terms of the geometric regression is .

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  • For ,

  • For ,

Now substituting  by their conjectures, we get,

Therefore we come to the conclusion that the formula involves the calculation of a geometric series and that difference between successive terms, say .

  1. Now we graph the table values.

(here)X-axis represents ‘n’ and Y-axis represents.

(here)X-axis represents ‘n’ and Y-axis represents .

(here)X-axis represents ‘n’ and Y-axis represents .(values in reference to ...

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