11. The Sierpiński triangle with its template consisting of a large square of side 1 and three squares of side ½

Orientation

In saying that a template completely defines a fractal we have glossed over the possibility that if the shape in the template has some symmetry then there will be more than one similarity that transforms the sh The Hйnon attractor with a portion enlarged to display its banded structurein6ape to a smaller copy. For a simple instance, there are 8 different similarities that transform a square to a smaller square. The scaling can be direct or there can be a rotation through 90°, 180°, or 270°, or a ‘mirror’ reflection in a vertical, horizontal, or one of the two diagonal lines, see Figure 13; we refer to these possibilities as the orientations of the scalings. Thus for the template used for the Sierpiński triangle in Figure 11 we have a choice of 8 similarity transformations that move the large square onto each of the 3 smaller squares, so there are 8 Ч 8 Ч 8 = 512 different ways of choosing the three similarity transformations. Figure 14 shows two of the possible attractors resulting from the same template but with different choices of orientation for the similar copies. Note that the attractor of Figure 14(a) may be obtained from 8 different combinations of orientation of the similarities: the portion of the attractor in each of the three small squares may be obtained from the whole attractor either with a rotation or by a reflection in a diagonal. On the other hand, there is only one choice of orientation for each of the three similarities that gives the attractor of Figure 14(b).

12. Self-similar fractals with their templates: (a) a snowflake, (b) a spiral

13. The 8 symmetries of the square indicated by the positions of the face

It turns out that there are 456 different attractors defined by this ‘three squares’ template. Of these, 8 have mirror symmetry about the rising diagonal, and all of these result from 8 different combinations of orientations for the three small images. The other 448 result from a unique combination of orientations.

Thus if the shapes in a template shape have some symmetry, the template alone need not define a unique attractor; however, once the orientations of the similarity transformations are specified, there ceases to be any ambiguity. This important fact may be stated as follows:

14. Self-similar fractals with the same template but with different orientations of the similar copies

Given a template, there exists a unique set (called the attractor and usually a fractal) that is made up of similar copies of itself which are scaled and positioned according to the shapes in the template, provided that the orientation (i.e. reflections or rotations) of each copy is specified.

(The perceptive reader may realize that this is still not strictly true—there are the powers are not whole. pe ‘pathological’ sets which also fit the templates: for example the ‘empty set’, which has a blank picture, fits any template. But for here this statement is good enough.)

Templates and functions

Functions were introduced as a rule or formula that takes each point of the plane and ‘moves’ it to a new position. There is another way of thinking of functions which emphasizes their geometrical effect. A (black and white) picture in the plane is made up of a large collection of points (these might be thought of as the dots or pixels on a computer screen that make up the picture). A function moves each individual point to a new point. Thus the set of points that make up the picture are moved by the function to a set of new positions, which taken together form a new picture. The function transforms the original set or picture into a new one, Figure 15. The word transformation is used synonymously with ‘function’, particularly when thinking about the geometrical effect on objects. (The words map or mapping are also used—in the familiar sense of the word, a map is really a function that associates each point on the ground with a point on the page of an atlas, at the same time transforming a coastline, say, to a corresponding wiggly curve on the page.)

We have encountered the function which moves each point in the plane to the point midway between it and the origin. Its geometrical effect is to shrink everything down towards the origin by a factor Ѕ. This function will transform any picture to a Ѕ size copy—thus it represents a similarity transformation of scale Ѕ.