Space-filling curve

The E curve belongs to the group of so-called FASS curves ( space- filling, self-avoiding, simple, self- similar ). These curves are space or area filling, selbstausweichend (ie center be - and non -free), simple and self-similar.

By repeating their design process in which each segment of the polyline is replaced by a reduced image of the entire polyline, the E curve does not need to cut any point of a square area arbitrarily close itself after a sufficient number of construction steps. The limit of this infinite sequence of self-similar curves fills the area completely.

Explanation of the design process

A Square is an e- curve of level n = 1 inscribed:

In order to view easier, as each segment of the polyline to be replaced in the sequel, the following pattern is used in which on the one hand, a distinction between light and dark squares and the other side by the orientation of the squares (see marking) must be observed:

Subsequently, the dark squares are replaced by the same pattern ( note orientation! )

And the bright squares ( note orientation! ) By the negative image of the pattern:

After this step, one obtains a curve E of stage n = 2:

This procedure can now be continued indefinitely. Here is an illustration of the level n = 3:

And a representation of the stages n = 1,2,3 in a common image: