15. A function moves points to points and so transforms sets of points to sets of points

Templates are simply a diagrammatic way of expressing a family of transformations or functions. Each of the smaller shapes codes the similarity transformation which takes the large shape and scales it down to the size and position of the smaller one. Each point inside the large shape is moved to the corresponding position within the smaller one, so any picture inside the large shape is transformed to a similar picture inside the smaller shape.

The Sierpiński triangle of Figure 11 has a template comprising the ‘unit square’, that is the square of side length 1 with corner with coordinates (0, 0), (1, 0), (0, 1), (1, 1), and three smaller squares of side-lengths Ѕ. The three functions giving the three similarity transformations that transform the unit square to the bottom left, bottom right, and top left squares respectively, have the formulae:

The first function just shrinks everything down towards the origin by a scale factor Ѕ. The second function does the same thing and then, because of the ‘+Ѕ’ term, shifts everything a distance Ѕ to the right. The third function scales by a factor Ѕ and then shifts a distance Ѕ up. As well as transforming the square, the functions transform the fractal itself. Thus (x, y) → ( 0.094 + 0.123i8941½ x, ½ y) shrinks the entire Sierpiński triangle onto the scale ½ copy inside the bottom left square. Similarly, the other two functions in (A) transform the entire Sierpiński triangle onto the similar copies in the bottom right and top left small squares. Together, these three scale ½ copies make up the entire Sierpiński triangle. The three functions (1) describe the basic self-similarities of the Sierpiński triangle—they are a mathematical way of describing the template. The Sierpiński triangle is completely defined by the template and orientations, or equivalently by the three functions. However, the functions have the advantage that they also encode the orientations of the transformations—these do not have to be noted separately. For example, the three functions that transform the fractal in Figure 14(b) to the similar copies in the three smaller squares are with the second function differing from that in (1).

We saw that a template (with given orientations) defines a unique fractal. This fact can be expressed in the language of transformations or functions, namely that a family of transformations defines a unique object.

Given a family of contracting transformations, there exists a unique set (called the attractor and usually a fractal) such that, taken together, the transformed copies of the set make up the entire set.

This ‘theorem’ or ‘mathematical fact’ can be stated and proved in a formal mathematical way. Note that the theorem states two things: that such a set exists, and that it is unique—there is only one figure which satisfies the stated property. This is typical of many major theorems in mathematics which guarantee the existence and uniqueness of entities with certain properties. Notice also that this statement does not require the transformations to be similarities—it holds for any contracting transformations, that is transformations that shrink sets in some way, and similarities of scale less than 1 are a special case of this. A family of contracting transformations is called an iterated function system, and the theorem is sometimes called the ‘Fundamental Theorem of Iterated Function Systems’. In these terms the theorem simply asserts that an iterated function system has a unique attractor.

Drawing fractals

Given a template or a family of transformations, how can we obtain good pictures, usually on a computer, of the unique fractal so defined? One widely-applicable method was indicated above for the von Koch curve in Figure 10 where we repeatedly substituted scaled down copies of the template into itself to get increasingly good approximations to the fractal.

An alternative method that is easy and efficient to program, called the Chaos Game, was introduced by Michael Barnsley. This produces a sequence of points in the plane (i.e. on a computer screen) that give a very good approximation to the fractal. This is rather like drawing fractals by iterating a function as we did for the Hйnon attractor, the difference here is that at each step the function to be applied is chosen at random.

Let’s consider this for the is lost by assuming that thes shapesSierpiński triangle with its template expressed by the three functions in (1). Take any initial starting point, the origin (0, 0) will do. Select one of the three functions at random, for example by tossing a die, and if it comes up 1 or 2 choose the first function, for 3 or 4 choose the second, and for 5 or 6 choose the third, and apply the chosen function to the initial point to get a second point. Throw the die again to select another function and apply it to this second point to get a third. Continuing in this way, we get a sequence of points, each obtained from the previous one by applying one of the functions chosen at random. These points will not be arbitrarily scattered across the picture but will jump around the Sierpiński triangle and after a time will fill out the whole fractal. Omitting the first 100 points (to allow the procedure to settle down) and plotting the next 10,000 points will give a good computer image of the Sierpiński triangle. This method is effective for computing a picture of the fractal defined by any family of contracting transformations. The chaos game was used to draw the fractals in Figure 12.

Self-affine fractals

An affine transformation is more general than a similarity, in that it scales by different ratios in different directions and so elongates objects, for example affine transformations transform squares into rectangles or parallelograms, and circles into ellipses. A self-affine fractal is one that is made up of smaller affine copies of itself. Self-affine fractals can be represented by templates, but now the smaller shapes are affine copies, rather than similar copies of the large shape. Typically the template will consist of a large square and smaller rectangles or even parallelograms. Figure 16 shows a self-affine fractal with its template with each of the three rectangles containing an affine copy of the whole. For self-affine fractals the component parts may be elongated in certain directions, and this allows a much greater range of fractals to be constructed than just self-similar ones. Figure 17 shows a self-affine fern and a self-affine tree, which on looking closely are each made up of 5 smaller affine copies—they each have templates consisting of just 5 rectangles or parallelograms (the templates have not been shown to avoid the pictures becoming too cluttered).