Peano curve

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

It is defined as the limit value of a series of curves that can be constructed step by step.

In the two-dimensional case, an example of a Peano curve is the following: You start by dividing a square into nine squares of equal size, which are traversed in an S-curve. In the next step, each of these squares is subdivided again and the resulting squares are run through in S-curves, which are linked together as a new curve:

Scaling the curves to the same size results in the first four steps:

If one continues this method of recursion, one obtains a sequence of curves which converge point by point .

The limit value obtained is the Peano curve, on which every point of the starting square lies and which is infinitely long.

This procedure can easily be generalized to higher dimensions. A continuous surjective mapping (with ) also provides continuous and surjective mappings , and by concatenation one obtains a continuous surjection for every natural number . ${\ displaystyle f \ colon I \ to I ^ {2}}$ ${\ displaystyle I = [0,1]}$ ${\ displaystyle f \ times \ operatorname {id} _ {I ^ {n-1}} \ colon I ^ {n} \ to I ^ {n + 1}}$ ${\ displaystyle I \ to I ^ {n}}$ ${\ displaystyle n}$

More Peano curves

There is also another area-filling curve known as the "Peano curve". Their structure corresponds to the Cantor diagonalization . A section between two points is replaced by the structure of the first stage.


First stage Peano curve	Second stage Peano curve

literature

Giuseppe Peano: Sur une courbe, qui remplit tout une aire plane. In: Mathematische Annalen 36 (1890), pp. 157-160.

Web links

Commons : Peano curve - album with pictures, videos and audio files