LiveGraphics3D Example:
The Space-Filling Peano Curve

First Step

This example was created with the help of Robert M. Dickau's Mathematica code for L-systems, available on his homepage.

Space filling curves are also discussed in Stan Wagon's book "Mathematica in Action".

Second Step

This graphic was constructed by replacing each of the lines of the first step by several smaller lines.

Third Step

The curve "covers" a greater volume after each step. In the limit of infinitely many steps of replacements the curve in fact becomes "space-filling". (This is clearly impossible, if all curves are considered to be one-dimensional.)

Martin Kraus, April 15, 2021