The Dragon Curve

The algorithm for generating the curve can be succinctly described by an L-system (for Lindenmayer). A short course on L-systems may be found at

The L-system for this curve is

             Angle 8     
             Axiom FX    
             F=          
             Y=+FX--FY+  
             X=-FX++FY-

The symbols X and Y have no graphic meaning; they are used simply to define the rules for generating the curve. The first generation of the curve is the axiom FX, which means we, after ignoring the X, simply draw a line segment one unit forward. To obtain future generations, we apply the replacement rules for F, X and Y to the previous generation. Note that the F's in the previous generation are all deleted in the next generation.

   Gen. 1:                    FX
   Gen. 2:                 -FX++FY-
   Gen. 3:           --FX++FY-+++FX--FY+-   
   Gen. 4: ---FX++FY-+++FX--FY+-+++-FX++FY---+FX--FY++-         

Thus, on the fourth generation we draw 8 line segments of unit length. As we proceed to the next generation, we assume we scale down the length of the line segments so that the picture has the same size as before.

We now show the pictures of generations 2 through 4.

Generation 2:

Above we see a 45 degree turn clockwise (-) followed by a 90 degree turn (++) counterclockwise. This makes a 90 degree corner shape. We show the previous generation as a dotted blue path.

Generation 3:

Each line segment in the previous generation is replaced by a corner alternately pointing to the left or to the right.

Generation 4:

Finally, we can carry this out to generation 18 and see approximately what the limiting shape is.

The dragon curve fills out an "island" of positive area with a fractal boundary, i.e. the boundary is so oscillatory that its "dimension" is strictly greater than 1 (not a linear object).