# Math Portfolio - The Koch snowflake investigation.

The Koch Snowflake

The Koch snowflake (also known as the Koch star and Koch island[1]) is a   and one of the earliest  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   .

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, .

(5 s.f)

Number of sides

Initially there is an equilateral triangle at stage n=0, each of those sides is divided into 3 sides. And there is another equilateral triangle created at the center points. Thus from that I can conclude that each side becomes from sides.

as the diagrams are given to us, for the first 4 stages I counted the sides. There is a pattern present, and from that I observed that the number of sides increases by a factor 4 of the previous stage.

From the table above, I can determine the nth term as:

From the graph above, we can observe that the number of sides increases as n increases.

Length of sides

At stage n=0, there is an equilateral triangle which has 3 sides. Those 3 sides each ...