Koch’s Curve

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:

  1. Begin with a straight line (the blue segment in the top figure).
  2. Draw an equilateral triangle with the middle segment as base.
    Remove the middle segment
  3. Now repeat, taking each of the four resulting segments, dividing them into three equal parts and replacing each of the middle segments by two sides of an equilateral triangle (the red segments in the bottom figure).
  4. Continue this construction.
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.

 

0

1

2

3

4

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:
1. Recall that the line segment in Stage 0 was 1 meter long. Follow the process and compute the length of Stage 1, remember that each straight segment is the same length.
2. Compute the lengths of the next few stages (you made need a calculator for this). Can you see a pattern? Find a formula for the length of Stage n. Check your formula against those you previously computed.
3. What happens to the lengths as n becomes very large? Do the lengths settle down to a particular number? How are they behaving? Your answer should make you feel a little uneasy if you’ve never done this before.
Don’t panic. There are three possible answers to the command: “Find the limit of this sequence (of numbers).” The limit exists and is finite. The limit exists and is infinite. The limit does not exist. Notice that existence is an important part of the answer…

 

Online calculation of the Koch Curve:

http://www.arcytech.org/java/fractals/koch.shtml

Basic Concepts in Nonlinear Dynamics and Chaos:

http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Workshop.html

Other Sources:

http://fractalfoundation.org/resources/fractivities/koch-curve/

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