As we have stated earlier, one of the original goals of introducing
*L*-systems was to model growth of various kinds. To do that, we
should attach some natural meaning to the formal symbols which are
manipulated in the *L*-system. In this section, we will begin to see
how the production rules can have geometric interpretations; the
resulting *L*-systems can produce wonderful geometric objects: various
kinds of *fractal* curves, shapes that mimic the natural world,
and tilings of various kinds.

We will start with a very simple *L*-system:

The generations simply double each time: *a*, *ab*, *abab*,
*abababab*, etc. To *a* and *b*, we associate the shapes:

The solid black lines indicate the curve that is drawn. The dotted
blue line is for reference. The triangle is an isosceles right
triangle. The productions in our *L*-system now represent a change in
the geometry of these configurations. The solid black lines become
dotted blue lines, while a solid black curve is drawn along
smaller isosceles right triangles along the dotted blue lines.
The two new configurations are

**Figure 4:** Production Rules for Dragon Curve

The first four generations of this *L*-system are displayed below,
starting with the *a* triangle.

**Figure 5:** First Four Generations of Dragon Curve

As we carry out more and more generations, a wildly kinky curve
known as the *Dragon* appears.

**Figure 6:** Dragon Curve Generation 10

We may start to see some beautiful emergent behavior in this dynamical
system. The curve apparently loops back to close off many small
squares. These squares cluster in a sequence of ``islands,'' each
island connected to the next by a single strand. The limiting shape is
the *dragon fractal*:

Mandelbrot [Man83] calls this the Harter-Heighway dragon; an extensive discussion of its properties can be found in [Gar67, DK70].

There is a natural dynamical process of ``folding'' at work in the creation of the dragon curves. Start with a rectangular piece of paper which we shall view from the edge. Fold the right half over the left half, with a sharp crease down the middle. Take the folded paper and fold again the same. Continue this folding process for a few more generations. The appearance of the edge is shown on the left side of the figure below. After a number of folds, unfold the paper, and spread each fold to an angle of exactly . The resulting edge curve is our dragon. The right half of the figure shows the results for the first few generations.

There is another natural encoding of the dragons that we can see in
the pictures. Following the curve from beginning to end, each turn is
either to the left or to the right. Thus, each generation of the
dragon corresponds to a sequences of *L*'s (lefts) and *R*'s (rights).
In the next picture, we show generation 4 with all the turns labelled
as *L* or *R*.

**Figure 9:** Generation 4 labelled with left and right turns

The corresponding sequences up to generation 4 are listed below.

1 fold | L |

2 folds | LLR |

3 folds | LLRLLRR |

4 folds | LLRLLRRLLLRRLRR |

There is a vague similarity to the Thue-Morse sequence. Indeed, there
is a relationship, but it is fairly elaborate to work out; see
[DFv82] for more details. The relation with
the paperfolding process suggest many interesting variations on the
dragon curve. For instance, instead of always folding the right half
over the left, we may occasionally fold the left half over the right.
Depending on the sequence of paperfolding instructions we get
different dragon curves. Strictly alternating left and right folds
produces a sequence of *L*'s and *R*'s that arose in a question of
analysis (about trigonometric sums) first considered by Rudin and
Shapiro (see [Rud59]).

Mon Aug 19 17:21:15 CDT 1996