2 - The Koch curve

    a - Niels Fabian Helge von Koch

Born: 25 Jan 1870 in Stockholm, Sweden
Died: 11 March 1924 in Stockholm, Sweden

Niels Fabian Helge Von Koch

Helge Koch was a student of Mittag-Leffler and succeeded him in 1911 at Stockholm University. He is famous for the Koch curve. This is constructed by dividing a line into three equal parts and replacing the middle segment by the other two sides of an equilateral triangle constructed on the middle segment. Repeat on each of the (now 4) segments. Repeat indefinitely. It gives a continuous curve which is of infinite length and nowhere differentiable.
Koch's principal results were on infinitely many linear equations in infinitely many unknowns.
 

   b - The basis pattern

Let us have a look at what we are looking for. The Koch curve with 5 iterations (we will see next what this number of iteration is) looks like this:
Koch Curve
- Koch curve -
The Koch snowflake is an Iterated Function Systems. Iterate means to repeat an operation, generally using the last result of that operation as the input. An iterated function system then is a system where a function is repeated over and over again itself.
According to Benoît Mandelbrot, this curve is a coarse but sufficient model of coast.
As a matter of fact, a coast like the one of Britain, are fractals: the perimeter depends on the scale at which you look. Every time you look closer, you discover another refinement that increases the length. To measure the perimeter, you have to specify a scaling factor.

But let's focus back on our Koch curve. To construct the Koch curve, we have to find the elementary pattern:

We begin with a one single segment, which length is 1.

Then, you cut it into three other segments, which length are 1/3 for each one.
You draw an equilateral triangle to replace the 1/3 long middle segment.
This first step should not be very hard to program!
Knowing that, let us see how we can arrive at the Koch snowflake drawn upper.
 

   c - One more step toward complexity


Now that we have seen how to create the basis pattern, we have just to repeat it for each of the four segment replacing the first one!
This will be the Koch curve after two iterations. If you do it one more time, then you have done three iterations.

Pattern
- the basis pattern, the pattern after one and two iterations -

And now, we just have to apply this principle with an equilateral triangle as initial figure.
This is what we could obtain:

- evolution toward the snowflake -
Here is a small animation for the pleasure.
Aniamtion of the Koch curve
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