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.
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