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Math Portfolio - The Koch snowflake investigation.

Extracts from this document...

Introduction

The Koch Snowflake

image08.png

The Koch snowflake (also known as the Koch star and Koch island[1]) is a mathematicalcurve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry" by the SwedishmathematicianHelge von Koch.

image09.png

In the above stages, certain notations are used for the nth term:

Nn: number of sides

Ln: length of a single side

Pn: the perimeter

An: the area of the snowflake

The Koch’s snowflake curve, simply starts of an equilateral triangle which is when n=0. The triangle has 3 sides, which are 1 unit each. Then each of those sides is divided into 3 equal parts. Thus each of those 3 parts is 1/3 unit. On the middle part an equilateral triangle is drawn. And this continues in the following stages.

image15.pngimage00.png

Below are the values for the first 4 diagrams, image20.png.

        n

       Nn

       Ln

       Pn

       An

       0

       3

        1

        3

image19.png

       1

      12

image11.png

        4

0.57735

       2

      48

image12.png

image16.png

0.64150

       3

     192

image13.png

image17.png

0.67001

(5 s.f)

...read more.

Middle

image03.png

image04.png

image18.pngimage05.png

image06.png

From the graph above we can observe that we can see as the number of sides increase (n), the perimeter also increases.

Area

The area of the 1st stage at n=0 is image19.png, as the shape changes in the 2nd stage, the area increases to the area of the first triangle+ area of new triangles

  • A1 = A0 + area of new triangles

There is an area scale factor present, which is image12.png.

  • A1 = A0 + new triangles ×image12.png× A0

A1 = A0 (1 + new triangles ×image12.png)

At the 2nd stage the area scale factor becomes

image21.png= (image12.png×image12.png).

  • A2 = A1 + new area

The table below has the areas of the 1st four stages:

       n

      Nn

Area scale factor

Extra triangles

      0

       3

     -

    -

      1

      12  

image12.png

     3

      2

      48

image22.png

     12

      3

     192

image23.png

     48

A0: image19.png× 12 = image19.png

  : A0 (1 + new triangles ×image12.png)

    : image19.png (1 + 3 ×image12.png)

A1: image19.png (1 +image24.png)

Stage 2 has 12 new triangles and also the scale factor becomes square of the previous scale factor. So it is image22.png

A2: A1+ new area

   : A1 + (12 ×image21.png×image19.png)

   : image19.png (1 + image11.png) + 12 ×image21.png×image19.png

 A2: image19.png (1 + image11.png+ image25.png)

The formula can be generalized here:

An: image19.png (1 + (image24.png +image26.png+image27.png+… …+ image28.png)                                                    

image07.png

Area Graph

image29.png

...read more.

Conclusion

From the table we can observe that from the 17th term, the area of An+1 equals An to 6 decimal places.

Also from the table, we can see that at n=15, the area is 0.69281897 and as you go ahead, the consecutive terms start to have very minute changes and the difference isn’t a lot.

The 17th term: 0.69282006

       18th term:0. 69282020

The difference between the two only is 0.00000014, which is very minute. This shows that as the n increases and goes up to infinity the area increases very minutely (tends to 0).

Limits and Scope

The calculator and the ms excel provided acute values along with sufficient amount of significant figures.

The limitations could be that there is no possibility of finding the value for n= ∞

Bibliography

image45.png

http://en.wikipedia.org/wiki/Koch_snowflake - intro

http://library.thinkquest.org/26242/full/fm/images/51.gif - picture of snowflake

http://www.enotes.com/w/images/thumb/d/d9/KochFlake.svg/280px-KochFlake.svg.png - the four stages

http://www.emeraldinsight.com/content_images/fig/1560120405005.png - curves

...read more.

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