# The Koch Snowflake

The Koch Snowflake:

Description:

In 1904 the swedish mathematician Helge Von Koch created the Koch snowflake. The Koch snowflake is a fractal curve, a geometric shape which can be subdivided into various parts. This fractal curve is built starting with a normal equilateral triangle (stage 0) and then to get to the following stage the inner third of each side is removed and another equilateral triangle is built where the side was removed. This process can be repeated various times. The first 4 stages are shown below:

Stage 0                            Stage 1                             Stage 2                          Stage 3

The aim of this investigation is to find an expression linking the area of all stages.

Method:

Part 1:

We’ll start by creating a table that shows various values at each different stage

N n = the number of sides of each stage

L n = the length of a single side of each stage

P n = The length of the perimeter of each stage

A n = The area of the snowflake of each stage

(Assuming an initial side length of 1)

Area of Stage 0 =   base x height

2

Height (h) = cos -1 (½ / 1) = 60º = ½

= tan 60 () x  ½ =

Area =              =

2

Area of Stage 1 = Area of Stage 0 + Area of 3 shaded triangles

Area of shaded small triangle = base x height

...