## 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
The result of applying _{ }= A and this
set is called attractor of the Iterated Function System {R^{2 }
; w_{1},_{ }w_{2}}. In fact, this
set is the fixed point of the Hutchinson operator W .
Approximation of this set is shown in Figure
32 (left). This image is obtained by so-called 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 { _{ }is an
invertible mapping of the plane, then the IFS {R^{2 }
; f _{1},f_{2}}
where f =
_{i
}f^{-1}_{* }
w_{i }_{* } 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 {
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 his |