Introduction

It is best to understand self-similarity in its geometrical sense. However, before we discuss it in this way, let us examine M. C. Escher's square-limit which has been reproduced in Figure 3-1^3.1. The drawing is coded in a graphical language by first defining a very simple shape and then a set of operations to be applied to it. The progression of a fourth of this drawing is illustrated in Figure 3-2. When we look at the center of the piece in figure 3-1, we are more conscious of the lines and areas which create the shapes; in other words, we create a mental representation of how the shapes look to us. As we move toward the outer edges, the shapes start to turn into textures; thus, the same shape and the same procedures are used in two levels of our vision perception. The procedure which creates this drawing is a recursive process. It could be made as big as one would wish. However, what we see on the page is actually just a snapshot of the forth level of recursion, and actually what is coded in this document is not the exact drawing but just the procedure. Therefore if one had access to the machine readable format of this document, one could change the number of levels of recursion and create a picture with more or less detail.

The self-similarity of this drawing is a bit difficult to grasp. If the drawing was made so that the shapes were built around the edges and the recursion process filled the center of the page, we could take any carefully picked segment of the picture form its center and magnify it, and we would come up with the same picture. (Many of Escher's engraving and drawing have this property; a very clear example is Path of life II by M. C. Escher). In this drawing (figure 3-1) there are actually no defined edges. If we cut a carefully chosen square from the middle of the drawing and then stretch every other part toward the center of the drawing so that the cut square would disappear, we would again come up with the same picture, except that some of the gray scales would be different, in this case, we call this picture self-affine.

Schroeder opens his recent book called ``Fractals, Chaos, Power Laws'' with the following paragraph[43, page xii]:

The unifying concept underlying fractals, chaos, and power laws is self-similarity. Self-similarity, or invariance against changes in scale or size, is an attribute of many laws of nature and innumerable phenomena in the world around us. Self-similarity is, in fact one of the decisive symmetries that shape our universe and our efforts to comprehend it.

Invariancy against change of scale is called self-similarity, and if there are more than one scale factor involved we call that self-affine.

In this chapter, we will try to create an impressionistic view of what self-similarity is, and touch upon a few of the cases which create its history. Self-similarity is created when a self-referential entity is observed. Chaos provides a physical proof of the tangible importance of the idea of self-similarity. Self-referentiality is deep at the heart of Gödel's proof, whose real implications for mathematics and logic, we believe, is not yet fully understood.

Figure 3-1: Square Limit by M. C. Escher

Figure 3-2: Square Limit progression