Session: May 2015
Global Indian International School
Topic: The Koch Snowflake
Fractal Geometry and its Applications
To investigate the underlying mathematical patterns of the Koch Snowflake.
As an artist, the beauty of geometrical patterns has always fascinated me, and a research conducted to gain inspiration for a new painting made me stumble across the concept of fractals. I took a liking to this topic immediately and read further but could not satiate my curiosity by reading one or two articles and found myself pondering over the mysteries of this phenomenon a number of times after finishing the painting too. A particular kind of fractal called the Koch snowflake…show more content… Then the lines common to the old and new triangles are removed. This is termed as the first iteration. – (f) The steps described in (b) are repeated four times, and hence, (f) is the fifth iteration.
The Koch snowflake is formed when the number of iterations tends towards infinity. This results in the snowflake having the property of self-similarity, displayed by all types of fractals. Thus, the snowflake can be zoomed infinitely and will still display the same pattern (shown on figure 1 below)
Properties of the Koch Snowflake
This snowflake has unique properties, which are not exhibited by most other geometrical shapes.
What is especially striking about the Koch snowflake is that the shape has an infinite perimeter, but encloses a finite area. When I first read this, it did not seem plausible to me, so I decided to derive formulae for both the area and perimeter of the snowflake in order to prove this property.
In order to find the area of the infinitely iterated Koch snowflake, I started off by finding the equation for the area of an equilateral triangle with the side ‘S’ (Figure…show more content… In fact, the only knowledge I did have about dimension was that a simple geometric shape drawn on paper is two dimensional, while objects like spheres and cubes are three dimensional.
Through my research, I learned that the dimensions5 of an object can be formally defined as the minimum number of coordinates needed to specify any point within it. It can hence be deduced that the dimension of a line segment is one, that of a square drawn on paper is two and subsequently, that of a cube is three.
In contrast, fractals such as the Koch snowflake have fractal dimensions. Fractal dimensions6 allow us to measure the degree of complexity of objects or shapes by evaluating how fast our measurements increase or decrease as our scale becomes smaller or larger.
Dimensions can be calculated by a number of different methods and I chose the self-similarity dimension to determine the dimension of the Koch snowflake. I found this method particularly appropriate for the shape since the snowflake exhibits the property of self-similarity at every level of magnification.
The equation for the self-similar dimension