8. The figures shown are all similar to each other

9. The von Koch curve together with its defining template

A self-similar set is one that is made up of several smaller similar copies of itself. We’ve seen that the von Koch curve is self-similar since it comprises 4 suitably placed scale copies of itself. This may be represented diagrammatically by a template consisting of one large rectangle and 4 smaller ones, each a scale copy of the large one, see Figure 9. The part of the von Koch curve framed by each of the smaller rectangles is a scale copy of the whole curve framed by the large rectangle. The size and position of the rectangles specify the scaling and positioning needed to fit together the four smaller copies to make up the whole von Koch curve.

But much more than this is true. The template is a simple diagram made up of classical geometrical shapes, namely five rectangles—there is nothing fractal about it. Nevertheless, the template completely defines the fractal: the von Koch curve is (essentially) the only object that is made up of smaller scaled copies of itself positioned as indicated by the template. It is easy to reconstruct the von Koch curve from the template by repeated substitution of a scaled template into The Hйnon attractor with a portion enlarged to display its banded structurein6the smaller rectangles. The first couple of stages are indicated in Figure 10 from which it should be clear that the small rectangles form patterns that get progressively closer to the von Koch curve itself.

10. Reconstruction of the von Koch curve by repeated substitution of the template in itself

This is a particular case of a very general and powerful method of describing and constructing fractals. A template consists of some simple shape (square, rectangle, triangle, etc.) and a number of smaller similar copies of the shape positioned somehow in the plane. Each of the smaller shapes represents a similarity transformation or scaling transformation that takes the larger shape and scales and repositions it to coincide with the smaller copy, scaling and positioning any figure drawn inside correspondingly. A fundamental property is that templates define fractals. That is, given a template, there is essentially just one figure, usually a fractal, made up of smaller copies of itself scaled and positioned according to the shapes in the template. (This statement is not quite accurate, but we will be more precise soon.) This figure is sometimes called the attractor of the template.

For a template made up of similar shapes, the attractor will be self-similar. Figures 11 and 12 show several self-similar fractals with their templates. In each case note that what is seen in each of the smaller regions of the template is a scaled copy of the whole. Figure 11 is known as the (right-angled) Sierpiński triangle made up of 3 copies of itself at scale ½. The snowflake of Figure 12(a) is made up of 4 smaller copies of itself; 3 are at scale ј whilst the larger central copy is at scale and has been rotated by 180° (or alternatively reflected in a horizontal line). This example illustrates two variants: the scalings of the similarity transformations do not all have to be the same, and some of the copies may be reflected or rotated. The spiral in Figure 12(b) is remarkable in that it is defined by just 2 similarity transformations. The first scales down only slightly with a 19/20 ratio but with a 45° rotation, and the second transforms the whole picture to the spiral at the end of one of the arms of the main spiral scaling at 1/5.