This post has been imported from my old blog.
The base case of the construction is an equilateral triangle. For each successive iteration the construction proceeds as follows:
- Divide each edge of the polygon (say of length a) into three equal segments of length a/3.
- Replace the middle segment with an equilateral triangle of side a/3.
- Remove the base edge of the new equilateral triangle to form a continuous curve with other line segments.
Thus each edge is transformed as such:
The first 5 iterations look like:
Two properties of this fractal may seem paradoxical at first but are easily shown. I will show the derivations here for the interested without skipping too many steps.
For an infinite number of iterations the perimeter of the fractal (the length of all the edges) tends to infinity whilst the area enclosed remains finite.
First consider the number of edges for an infinite number of iterations. Initially the number of edges is 3, each iteration transforms a single edge into 4 edges (see above), thus the series of the edge count is: 3, 12, 48, …. The general form is:
Starting from an edge length of a each iteration divides the edge length by 3. The following defines the edge length for iteration n and from this calculates the perimeter for iteration n. The limit as n tends to infinity is found, showing that the
perimeter is infinite.
Thus it is easy to see that the perimeter length is infinite through standard results on limits.
The area calculation is a little bit more involved. The result can be seen on Mathworld or Wikipedia but a full derivation isn’t given, so I show my derivation here for the interested.
The area of the n-th iteration of the Koch snowflake is the area of the base triangle + the area of the new, smaller, triangles added to each edge.
From the above length calculations and construction we know that each iteration of the construction divides the length of the edges by three. First consider the relationship between the area of a triangle of edge length a and a triangle of edge length a/3
Unsurprisingly we see that dividing the edge length by 3 divides the area by 9. At each iteration we add new equilateral triangles to each edge, a ninth of the area of the previous iteration’s triangles.
We formulate this as a summation of base case A0 plus the next n-1 iterations where the triangle area at each stage is A0 divided by 32n or 9n (successive divisions of the area by 9). The number of edges is defined by the previous iterations number of edges 3 * 4n-1, thus at the n-th iteration we need to add this number of triangles to the snowflake.
Thus the area is finite at 8/5 times the area of the base equilateral triangle.
So the Koch snowflake construction has been introduced and it has been shown relatively easily that the area of a Koch snowflake tends to a finite limit of 8/5 times the base case area (5) and that the length of perimeter tends to infinity (4). Next time I will probably talk a bit about the fractal dimension of the Koch snowflake and also show an interesting problem that uses the Koch snowflake.
As a last note, following on from the blog post I made about the Python turtle library, a turtle program of the Koch snowflake for Python can be downloaded here.