The Sierpinski curve, invented by Waclaw Sierpinski, is a plane-filling, non-intersecting fractal. It has the rather odd properties of being infinitely long, enclosing every interior point in a square, yet covering only 5/12 of the area of said square.

It looks like a recursively subdivided "knot", like this one:

/\___/\   /\___/\
\     /   \     /
 |   |     |   |
 |   |     |   |
/ ___ \___/ ___ \
\/   \     /   \/
      |   |
      |   |
/\___/ ___ \___/\
\     /   \     /
 |   |     |   |
 |   |     |   | 
/ ___ \   / ___ \
\/   \/   \/   \/