Koch’s Curve is a basic Fractal. ‘Fractals’ where first described in 1975 by Benoït Mandelbrot, but those fascinating figures were already discovered 100 years earlier by mathematicians investigating bizarre mathematical behavior, and called ‘monster curves’.
A fractal is a geometric object which is highly irregular at every scale. Some of the most famous fractals have self-similar structure: they have a repeating structure at all level of magnitude. One of the most familiar examples is Sierpinski’s Triangle. Many of these fractals can be generated by repeating a pattern in a recursive process.
Lets start with a very early fractal-like phenomenon
In 1904 Helge von Koch introduced the Koch curve. Here is how the curve is recursively constructed:
|In the picture at right, suppose for the sake of argument that the line segment in Stage 0 of the figure is 1 meter long.
The next stage, Stage 1, is produced from the previous stage by first dividing the line in Stage 0 into three equal pieces of length 1/3 the original size, then removing the middle third and inserting the tent of an equilateral triangle.
Stage 2 is obtained from Stage 1 by applying the above process to each of the four straight line segments in Stage 1. And we continue… If you want to draw Stage n you simply apply the process to the previous stage, Stage n-1 . But, of course, you need to know all the stages prior in order to do this. The result is a sequence of drawings becoming more complex the higher the stage number, but still looking somewhat like the previous members of the sequence.
You can see in the figure that already at Stage 4, the drawing is quite complex with much detail. In fact, if you continued the construction further you might say that stages 4,5,6,7,… don’t look that much different from one another, and you’d be right. Of course they are different fundamentally, but at the scale we’ve drawn them, we can’t see much difference.
|Niels Fabian Helge von Koch (1870-1924) was a Swedish mathematician who first played with the figures we are discussing. He noticed that as the stages progressed, the figures seemed to “settle down” to a figure not that much different from that in Stage 4, as we’ve observed. He asked the question, “What happens to the figures if we continue the process indefinitely?” In other words, suppose you would let your computer keep on calculating at high speed this algoritm, could you tell the difference between Stage 1,000,000 and Stage 5,555,679, or further? Does this sequence have a limit? When zooming into this curve it would look about like this:|
|In fact, this sequence of drawings does have a limit, in a technical sense, and that limit is called “von Koch’s Curve.” What’s interesting, is that if you arrange 3 copies of the curve along the edges of an equilateral triangle, you get the figure at left. Now it’s clear why it’s also been referred to as the ‘snowflake fractal’
What is the length of von Koch’s curve? The only way to answer such a question is using limits. Here’s a guide:
Online calculation of the Koch Curve:
Basic Concepts in Nonlinear Dynamics and Chaos: