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Maths HL Kochs Snowflake

Extracts from this document...

Introduction

The Koch Snowflake

image19.gif

The koch snowflake is a fractal that was identified by Helge Von Koch in 1904. It is created by starting with an equilateral triangle, removing the centre third of each side and replacing it with another, smaller equilateral triangle.

Table 1.1: Table of Values For the Koch Snowflake

Stage

Nn

Ln

Pn

An

0

3

1

3

0.433

1

12

image00.png

4

0.57735

2

48

image01.png

image02.png

0.6415

3

192

image03.png

image04.png

0.67

Finding the values of N was relatively easy as this was just a process of counting the sides. It was discovered that the Number of sides was

...read more.

Middle

. The stage 0 was equal to image07.png

 as the side length in stage 0 is equal to 1. Every stage after that was equal to the area of the previous iteration, + (Area of an added triangle) x (Number of added Triangles.).

It should be noted that the graph for Pn is divergent. The perimeter has an infinite value when n approaches infinity.

The graph of An is convergent.  A has a finite value when n approaches infinity.

Generalizations for behaviour within Graphs.

General Formula for Nn:

n

Nn

0

3

1

12

2

48

3

192

N

3(4)n

...read more.

Conclusion

ng" alt="image08.png" />

The general formula for length is one third of the previous length. The general formula is then 1/3n.

Verifications

N = 0        image09.png

 = image10.png

 = 1

N = 1        image11.png

 = image00.png

N = 2        image12.png

N = 3        image13.png

General Formula for Pn

n

Pn

0

3

1

4

2

5image00.png

3

7image01.png

n

image14.png

The general formula for perimeter is equal to the Nn multiplied by the Ln. 3(4)n x image08.png

 . The general formula is thenimage14.png

.

Verifications

N = 0         image15.png

N = 1         image16.png

N = 2        image17.png

N = 3        image18.png

...read more.

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