Many biological forms are ``branched,'' ``fragmented,'' or ``cellular'' in appearance and growth. To allow branching to occur in the turtle graphics interpretation of L-systems, we use the symbols [ (``push'') and ] (``pop'') to enable the turtle to follow a branch for a time at the end of which it returns to the position where it started branching. We will begin with a simple branching where a main ``trunk'' shoots off one side branch:
Angle 10This can be viewed as replacing each straight edge by the following configuration:
Axiom F
F=F[+F]F
The change of angle and one step is taken with the pop and push operators. Scaling so that the main trunk remains the same size, we see the sequence of generations in the figure below.
Figure 13: Generations of branching
Even with this simple L system, one can visualize a twig of a bush branching recursively. Remarkably, very realistic images of biological growth may be modelled by L-systems that are not very much more complicated. In FRACTINT, the reader should examine the series PLANT01-PLANT11, BUSH, and LEAF1-2. These and many much more sophisticated plant modelling systems are discussed in much greater detail in [PL90].
To end this section, we give a picture of the BUSH (by Adrian Mariano) system. The model is a single branching, slightly asymmetrical and with slightly curved branches.
Angle 16The angle
Axiom ++++F
F=FF-[-F+F+F]+[+F-F-F]
allows for a gentle curving.
The main trunk exists only before the branching, not afterwards.
The four +'s in the axiom orient the structure vertically.
The generation 5 picture is in
this figure.
Figure: Bush L-system, generation 5
As we pass to later generations, the branches of the bush system cluster in a way to give a three-dimensional appearance to the bush. True three-dimensional L-systems can be achieved by replacing + and - by a set of six three-dimensional rotations, one pair for each axis of rotation among the x-, y- and z-axes.
