A space-filling curve is a curve whose zigzagging lines cover the entire space they occupy. They astound mathematicians because they seem as if they ought to be one-dimensional. After all, they are lines. However, as the curve reaches its limit, space filling curves occupy a two-dimensional space, becoming a completely filled figure.
The curves above were created using an L-system. An L-system is created by starting with an axiom, such as a line segment, and one or more production rules, which are statements such as “replace every line segment with a left turn, a line segment, a right turn, another segment…”. When this system is iterated several times, the result is often a complicated fractal curve. More.