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.