Take a moment and review some of the pictures arising from the L-systems we have considered so far. The limiting geometrical object is an example of what is generally called in dynamical systems theory an attractor. Generally, this means some physical situation that the system in some sense converges to. For some symbolic dynamical systems, the attractors are the infinite sequences produced. For the L-systems coupled with turtle graphics, the attractor may be the limiting curve or shape. To arrive at the limiting shape, we must rescale the generations so that the ``step size'' d is decreased by a suitable factor from generation to generation.
When we view the limiting shape (or close approximations), the evolution and the rescaling are often visually apparent. Consider first the dragon curve. In this figure, we show how the limiting shape can be split into two congruent shapes.
Figure 18: Splitting the dragon: The two halves of the dragon below
are marked as Parts A and B. The split is approximately sketched
in red.
The original isosceles right triangle is marked in green in this
figure. The two smaller halves are each constructed on a segment of
length 
times the length of the segment of the
original dragon. Also the two halves are congruent: one may lay one
exactly on top of the other by a 
rotation and a
translation. We see that the dragon fractal is composed of two
congruent pieces, each of which is scaled down by
from the original dragon. This is an example of self-similarity:
similarity of the whole object to smaller pieces of itself.
To many of the graphic L-systems listed in FRACTINT and elsewhere we
can associate two numbers: a ``number of congruent pieces'' that the
shape may be divided into, and a ``scaling factor'' that scales each
piece into the original object. For the dragon fractal, the number of
pieces is 2 and the scaling factor is 
. These numbers can
partly be deduced from the L-system rules, although a universal
method for doing this is not known. For the dragon, we see in the
L-system production rules that each segment reproduces two smaller
segments of length times the original.
As a second example, consider the Koch curve, which we show below splitting into similar parts. The production rule converts each segment into 4 new ones of length equal to 1/3 the original length. Thus, it is not too surprising to see four similar pieces each one-third the size of the original Koch curve. In the exercises, we ask the reader to find the number of congruent pieces and the scaling factor for a number of the L-systems in FRACTINT.
Figure 19: Splitting the Koch curve
Self-similarity is a key concept in the theory of dynamical systems and the resulting fractal geometry. We must spend a moment to describe the language in which we may precisely talk about the concepts of congruence and similarity. We shall take up this task in the next section where we wish to concentrate on particularly symmetric L-systems and the resulting geometry.