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4. Conclusion: Can we handle fractals?

If our conclusion is contained in the statement that fractals are not art by themselves, but can be used to create art, in the similar way as looking in clouds, old wall or frottages, then there is a question: Can we model fractals so to put them in different positions, to rotate of stretch, to make  their symmetric images, to combine different parts of fractal sets freely, as we do with collages? Unfortunately, the answer is not so encouraging. It can be done in a very limited amounts. But, on the other side, this answer is not so unexpected. Modeling chaos is not simple. Putting a chaotic process to "draw" a picture that we want  is even more difficult. To give better explanation, we will use the easiest way of fractal construction: IFS theory. Moreover, we will choose the simplest type of IFS-s (Iterated Function Systems), this one containing affine mappings in the plane. In order to illustrate what we want to say, we will take two affine mappings of the plane R into itself [Dubuc]

w1(x, y) = (0.824074 x + 0.281482 y -1.882290,  -0.212346 x+ 0.864198 y -0.110607),

w2(x, y)= (0.088272 x + 0.520988 y + 0.785360, -0.463889 x - 0.377778 y+ 8.095795).

The result of applying  w1 and w2 to the unit square  A0 is illustrated in Figure 31 (left). Both mappings are contractive which is evident from spectral norms of square matrices being involved in w1 and w2 are 0.919475 and 0.74804 respectively. The corresponding Hutchinson operator  W = w1 U w2  maps A0 into A1 = w1(A0) U w2(A0). Iterating this procedure (Fig. 31, right) will converge to a compact set A in Ñ2. Thus, lim  An A and this set is called attractor of the Iterated Function System {R ; w1, w2}. In fact, this set is the fixed point of the Hutchinson operator  W .

Figure 31. Left. Two affine transformations for a "Seahorse" fractal and their fixed points f1 and f2; right. First four iterations of Hutchinson operator for "Seahorse" IFS.

Approximation of this set  is shown in Figure 32 (left). This image is obtained by so-called random algorithm [Barn] with 105  points. We call it "seahorse" and it falls to "dragon" family [Dubuc].

First thing that we see is proportion of the "seahorse" which is (approx.) 0.837934. Can we have some other proportion? Can we, for instance, obtain a "golden" proportion seahorse"? Of course, simple stretching of the image is out of question. What we need is a new IFS, say {R1,f2}with the attractor shown in Figure 32 (right). From [Barn] it is known that if  f i  is an invertible mapping of  the plane, then the IFS {R1,f2} where i = f-1 wi f  and where 

is the linear transformation that transforms the initial "seahorse" into "golden" one (Fig. 32). The asterisk here means the usual functional composition. If we replace the matrix in above expression with


the IFS {R21,f2} will produce the mirror-symmetric attractor (w.r.t. y-axis). Both attractors are shown in Figure 33 (left). Including the translation component for the vector [-10, 0]T, gives effect displayed in the same figure (right). Figure 34 shows couples of attractors in position of central symmetry and similitude perspective arrangement. Including more attractors in position of rotational symmetry brings figures as it is displayed in Figure 35. Note that these figures are produced by a set of  IFS's with the number of  IFS's, equal to the number of  visible "seahorses".

Figure 32. Left. "Seahorse" fractal obtained by Random algorithm; right. The same fractal as a golden  figure (in proportion  0.61822).

Figure 33.  The "seahorse" fractal in two mirror symmetric situations.

Figure 34. The "seahorse" fractal in a  point symmetric situation (left) and in perspective view. 

Figure 35. The "seahorse" fractal and two rotational symmetries.

Beside these simple geometric transformations, that preserves "fractality" there are also other features that can be applied in art constructions. One is self-affinity of the attractor. In fact the attractor is composed out of smaller affine copies of it. In other words, one can tile the attractor in as many sub-regions as it is the number of mappings in the IFS. In this way, it is possible to define infinitely many fractal tessellations of plane. Also, one can use different colors in rendering parts of the attractor. It can be applied in different adornments and mosaics.

We will conclude this article with the impression that fractals might be a useful tool that can be used as blotting to discover worlds of beauty newer seen before. Also, it can be used as a special kind of “frottage” that presented to the observer leave possible associations to him. In this sense we agree with Kerry Mitchell who 1999 proposed in hisFractal Art Manifesto, that Fractal Art cannot be fully computerized and random in the sense of stochastic and unpredictable. Instead, as he says, it must be expressive, creative and requiring input, effort and intelligence.