Peano Curve

a continuous curve in the Jordan sense that entirely fills a square—that is, the curve passes through all the points of the square. The first example of a curve possessing this

Figure 1

property was constructed by G. Peano in 1890, and a simple example of a Peano curve was given by D. Hilbert in 1891. The initial steps of Hilbert’s construction are illustrated in Figure 1.

The limiting curve obtained by continuing the construction ad infinitum will be a Peano curve that passes through all the points of the square D.