- Level: International Baccalaureate
- Subject: Maths
- Word count: 1084
Math Portfolio - The Koch snowflake investigation.
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Introduction
The Koch Snowflake
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.
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.
Below are the values for the first 4 diagrams, .
n | Nn | Ln | Pn | An |
0 | 3 | 1 | 3 | |
1 | 12 | 4 | 0.57735 | |
2 | 48 | 0.64150 | ||
3 | 192 | 0.67001 |
(5 s.f)
Middle
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 , 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 .
- A1 = A0 + new triangles ×× A0
A1 = A0 (1 + new triangles ×)
At the 2nd stage the area scale factor becomes
= (×).
- 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 | 3 | |
2 | 48 | 12 | |
3 | 192 | 48 |
A0: × 12 =
: A0 (1 + new triangles ×)
: (1 + 3 ×)
A1: (1 +)
Stage 2 has 12 new triangles and also the scale factor becomes square of the previous scale factor. So it is
A2: A1+ new area
: A1 + (12 ××)
: (1 + ) + 12 ××
A2: (1 + + )
The formula can be generalized here:
An: (1 + ( +++… …+ )
Area Graph
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
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
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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