Self-similarity and sets of affine maps

 

One theme throughout the geometrical examples we have seen so far is self-similarity. The objects can be dissected into pieces which are similar to the whole object. For example, consider the Koch curve. We have already seen how this fractal splits into four pieces each of which is similar to the original with similarity constant 1/3. Let K denote the whole Koch curve. Then the four pieces are , j=1,2,3,4, where the are the precise similarities. In terms of complex numbers (see this section for a discussion of complex numbers) we may write

for j=1,2,3,4. Here, and stand for complex constants.

We can figure out the similarities by a little complex number algebra. Figure 40 shows the pieces of the Koch curve.

  
Figure 40: Linear mappings and the pieces of the Koch curve

Knowing the location of the images of two points allows us to determine and in . From the picture, we arrive at the following information:

We can solve these equations to find and . Another way to do this is to observe that each piece is 1/3 as large as the whole piece K. Thus, the scaling factor is ; to find we just need to account for any rotation. The constant is always the image of 0. Either way, we may find the following to be true.

Smaller pieces of the Koch curve are obtained by combining several of the transformations. If each is one-third the size of K, then the sets are one-ninth the size of K. K is made up of all the pieces (briefer notation for ) for all possible pairs of indices i,j chosen from . In Figure 41, we show the position of each piece . Smaller and smaller pieces are obtained by applying more and more 's, for example:

where each a is chosen from . This piece has diameter equal to times the diameter of K. As , these smaller and smaller subsets of K converge to a single point of the Koch curve. Thus, each sequence in the code-space of (see this section for our earlier discussion of code-space)

corresponds to a single point in the Koch curve. We mark the location of in Figure 41.

  
Figure 41: Locations of

Some points of the Koch curve correspond to two different elements of code-space. That is, there are two sequences with . This is an interesting exercise for the reader to work out. In general, most points correspond to exactly one sequence in code-space, while countably many correspond to two different ones.

Many of the examples throughout our notes have an underlying set of transformations. As a second example, the Cantor set is composed of two pieces each of which is one-third the size of the original piece. There are then two transformations , given by the formulas

The first transformation maps I=[0,1] onto , and the second onto . In this case, there is an exact one-to-one correspondence between the sequences in the code-space of and the points of the Cantor set. If we have a sequence with each or 2, then for any , we may define , , , etc. Then our observations amount to

The particular ``seed point'' does not matter as long as it's in I. Surprisingly, the last condition is unnecessary; we shall see later that any seed point may be chosen in and the limit point will be the same.

The sphinx tiling (see this section) also has a set of four linear transformations corresponding to each piece that the sphinx tile is subdivided into. In the exercises, we pose the problem of identifying the code sequence of some of the tiles.

To end this section, we advance a beginning definition of Iterated Function System, or IFS for short (see p. 80, [Bar93]).

Definition: An iterated function system is a finite set of affine linear transformations of .

Here we do not require the w's to be similarities, but in order to discuss the attractor we will need to impose some condition on the w's. For now, let's recall that each w is determined by six real numbers

The notion of IFS applies to higher-dimensional spaces , and even more general situations, but we will limit our discussion to two-dimensional systems.