Peano curve

The Peano curve (after Giuseppe Peano ) is a space-filling curve ( FASS - curve).

It is defined as the limit of a sequence of graphs which can be gradually built.

In the two-dimensional case is an example of a Peanokurve is the following: One starts with the division of a square into nine equal squares, which are traversed in an S- curve. In the next step, each of these squares is divided again and run through the squares resulting in S- curves that are hung together as a new curve:

Scales the curves to the same size, is obtained as the first four steps:

Substituting this method, the recursion continues, we obtain a series of curves that converges pointwise.

The limit we obtain the Peano curve on which each point is located in the main square and is infinitely long.

This method can be easily generalized to higher dimensions. Also provides a continuous surjective map ( with ) turn steady and surjective illustrations, and by chaining obtains a continuous surjection for every natural number.

More Peano curves

There is also a further, surface -filling curve, which is known as Peano curve. Its structure corresponds to the Cantor diagonalization. Here, a distance between two points is replaced by the formation of the first stage.