Relationships with n
N against n:
As the number of sides increase with the rise in, each stage will have a number of sides which will be four times higher then the prior stage. With this information we can insinuate that the formula for the number of sides at a certain stage n is:
l against n:
in the chart above we are able conjure that the length of a single side diminishes with the progression of the snowflake or to be more specific, the length of one single side in a particular stage equals to precisely one-third of the length in the prior stage. Thus the concluded formula for the length of a single side in stage n:
P against n:
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 =
After cionsulting the prevous results in my graph I have proved that my results are correct.. Here is a diagram of one of the three original sides of the triangle in the fourth stage:
B) Supposing that n=6, I will determine the side, number of sides, its length, perimeter, and area at stage six:
A6= +x
→
A6= A6= + =
Successive values
Scrutinizing the patterns throughout the changed stages I was able to create a general formula for the area of a snowflake An.
When I place the formula of the area of a snowflake into a geometric sequence I can recognize a relation.
Common ratio r =
d = x 1 → d =
Conclusion
After soling the formula for An, now I am able to continue my research in the findings of the formulas for both the perimeter and the area of the Koch snowflake as 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.
