2 - The Koch curve
a - Niels
Fabian Helge von Koch
Born: 25 Jan 1870 in Stockholm, Sweden
Died: 11 March 1924 in Stockholm, Sweden
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
- 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
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.
- 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
This will be the Koch curve after two iterations. If
you do it one more time, then you have done three iterations.
- 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.
- evolution toward the snowflake -
Here is a small animation for the pleasure.
This is what we could obtain: