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# Koch Snowflake

Extracts from this document...

Introduction

MATH  SL   PORTFOLIO

“Koch Snowflake”

Aim

e

The aims of Koch snowflake investigation is to completely examine and understand the alterations in the number and lengths of sides, and further more the area and perimeter, as the snowflake undergoes different stages while in addition to also exemplify each stage and its complications. The fractal is created by starting with an equilateral triangle and removing the inner third of each side, what makes another equilateral triangle in its place, where the side has been removed.

To comprehend the data I decided to represent it in a table. As each stages changes, n  will represent each stage number, so at stage 0, n = 0 and so on.

 n Nn ln Pn An 0 3 1 3 1 12 1/3 4 2 48 1/9 16/3 3 192 1/27 64/9

From this table I observed that N, l, and P operate as geometric series. The value of N, the first term is 3 and the common ratio is 4. In l

Middle

Perimeter of the snowflake will increase as stages increase. The formula is actually the product of the length of a single side and the number of sides. This is how the formula for the perimeter at a stage n looks like:

A against n:

With every stage the area of the snowflake increases, but we are able to predict that at a certain stage the area will remain constant. This occurs when n is so big that draws near to zero, and because of this, it becomes indisputably irrelevant. The formula for the area of a snowflake at a certain stage n:

Investigating the formulas

A)  Assuming that   n=4,   examine the side, number of sides, its length, perimeter, and area at stage four:

A4 = +x →   =   =

A4 =+A4 =+

A4 =  = and because = 2187, then =

Conclusion

n becomes exceedingly large.

As n increases, the perimeter and area will also increase. The perimeter constantly increases and is continuously a product of the previous term and the ratio (4/3). Therefore, it will never be able to  reach zero. The area on the other hand does not  have a common ratio between successive terms. It has a configuration which works like a sequence,  which is noticeable within the formula:

We are able to recognidse the geometric sequence: .

As n increaseswill decrease in size. Or to be more specific it will approach zero, as the asymptote.

Asdevelops to be insignificant, we will get, which results to be a constant.

In this investigation I was able to not only recognize similarities and patterns within the snowflake and it stages by using information such as the perimeter, number of sides and the length of a single side, I was also able to conduct formulas and concluded that all formulas for this investigation were geometric sequence.

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