Many self-similar objects do exist in our world, the figure of a mountain or ridge line is often reflected in smaller formations of rock along its base. The bronchi of the human lung exhibit self-similarity over at least 15 levels. Fern frequently exhibit self similarity. Peaks and bends in the path of a long river are echoed in the sand formations along a single bank. It is suggested that the secret of encoding so much information in the DAN is in self-similarity.

A basic self similar object is the Von Koch curve or snow flake curve. Principle of the construction of this curve :

a) trace a line which you can cut in three equal parts of length a.

b) Erase the middle segment

c) Trace a first segment a at 60° and an other one next to it at 120°. Or create an equilateral triangle of length a in the middle segment erasing the base.

To construct the curve we start from an equilateral triangle instead of a line. The two first steps of the process give following pictures :

If we keep on doing the same process long enough we have a snow flake shape as shows the following picture :

Practically, on the computer the curve stops changing when the length of the elementary segment drops under the distance between two pixels, and we can stop the process. But mathematically we keep on going indefinatly. Which is going to create a curve of infinite length.

We can see that the shape is omnipresent in the picture, on every possible orientation and scale.

That is what is called self similarity.