Hilbert Curve Fractal

The Hilbert Curve is an example of a (linear) fractal. In particular it is a plane-filling curve, the first example of which was discovered by Giuseppe Peano in 1890. Plane-filling curve is an oxymoron along the lines of "pretty ugly" and "jumbo shrimp." This is because curves are one-dimensional, but planes are two-dimensional, so to have a curve that can completely fill part of a plane (i.e. become two-dimensional and yet remain a curve) is somewhat of a contradiction in terms.

It is called a linear fractal because at its final stage any small portion of it can be magified to become an exact copy of the whole. Fractals are created through a process of iteration, following a rule over and over. The first three stages of the creation of this curve are shown in the diagram above. The square is cut into 4 pieces, then 14, then 64. The curve is a path through the square that hits the middle point of each of the sub-divisions of the square. There is a specific process for the creation of this curve. The shape in the previous stage is copied 4 times, shrunk by a factor of 2, copied 4 times over - twice in the bottom with the same orientation as before and twice on the top, once rotated counter-clockwise and once rotated clockwise. The 4 copies are then connected by a line segment (shown above by dotted lines).

It's much easier to see than to say!!

These images were taken from the Plane Filling Curves page at the wonderful Cut the Knot site, which contains "interactive mathematics miscellany and puzzles." Speaking of INTERACTIVE, if you to the Plane Filling Curves page you can use a Java Applet to click and see the stages go from one to the other.