Computer Generated Images as Mathematical
Tools

** **

*Laboratory of
Applied Epistemology, University of Ghent*

*e-mail:
elizabeth.demol@ugent.be*

* *

* *

** **

**Abstract**

Since
the commercialisation of the computer, it became possible to visualise certain
aspects of mathematics that were not possible to visualise before because of
the complexity or the size of the datasets involved. Some of these computer generated images even have become the
icons of certain mathematical theories like for example fractal geometry. One
of the advantages of these visualisations is the fact that in using them,
certain properties that involve complexity can be *immediately* shown. This possibility will be discussed through
experiments done by the author.

** **

**1. Introduction**

*Everybody else took an exam in algebra and
complicated integrals, and I managed to take an exam in translation into
geometry, and in thinking in terms of geometric shapes.*

*(Mandelbrot in an interview)*

* *

The
computer created new possibilities for scientific research. One of these
possibilities is the visualisation of certain aspects of mathematics. These
computer generated images however are not only interesting *as* visualisations, but can also be used as instruments for doing
research. More specifically, they can help researchers to answer certain
structural questions i.e. questions concerning the large scale behaviour of a
given system and the structure that follows from this behaviour [See for
example 1, 2 and 3]. That these images can be used as research techniques is in
part due to their *directness* i.e. you
can gain knowledge of a systems’ structure just by looking at it. Especially
when large or complex systems are involved, visualising the structure of the
behaviour of a system can be a much more *direct
*and *quick* first answer to certain
specific questions. Of course this does not always imply that there is also a
rigorous explanation for the properties observed, but you at least know where
to start i.e. you know what the behaviour is. During recent research done by
the author, this method of using computer generated images as testing tools for
certain structural questions, was used [See 4]. It will be shown how this
method can work in practice, together with its advantages and disadvantages.

**2. Deriving fractals
from IFS-systems **

** **

One of the prototypical
examples of the use of computer generated images in mathematical research is
fractal geometry. There are many different methods to generate fractals,
depending on what kind of fractal one wants to be visualised [See 5]. In the
experiments 2-dimensional IFS attractors were generated through a variant of
the chaos game. An IFS-fractal is defined through a set of transformations *k*, which determine how a given point (*x*, *y*)
is transformed into another point (*x*_{1}*,**y*_{1}), where:* *

*
*

* x*_{1} = *aix* + *biy* + *ei* (*i* = 1, 2,...,*k*; {a,b,c,d}
є [-1, 1])

*y*_{1}_{ }= *cix** *+ *diy** *+ *fi *

* *

These transformations then define an attractor i.e. every
point in space will be attracted by the limit figure. Using such
transformations, one can generate fractal figures. One of the methods to do
this is the chaos game. The chaos game
is played as follows. Consider for example the following IFS-transformations:

*f _{1} (x*,

*f _{2} (x*,

*f _{3}
(x*,

We then have to choose an initial game point (*x _{1}*,

** **

* *

*Fig. 1 Image of the
Sierpinski triangle.*

Now
it must be remarked that when looking at the behaviour of an arbitrary point in
applying the chaos game on it, this point will have a unique orbit when
following it in its convergence to the attractor. On the other hand, it must
also be noticed that besides this, every point has its own convergence speed:
some points will need two iterations of the chaos game before reaching the
attractor, others three,...Now given these observations, the following question
can be posed: given the diversity of convergence speeds and individual orbits,
is there a general structuring principle behind this diversity? To answer this
question, one has to be able to compare the individual orbits and speeds of these
points in space, and thus find a way to observe the large scale behaviour of
one group of points in space, according to their orbits and speeds. To observe
how space is possibly structured according to these individual orbits, the
following experiment was set up. First of all a 3816 × 2319 array of points was
selected as being the points of which the behaviour was to be analysed. The
attractor is located somewhere at the centre of this array. Secondly, to each
point the same random string was assigned. This string consists of all the
numbers between 1 and the number of transformations which define the attractor.
For the Sierpinski triangle for example, this string consists of the numbers 1,
2 and 3. This string is then to be interpreted as the successive sequence of
transformations to be performed when playing the chaos game. Each point (*x, y*) is then transformed according to
this same sequence of mappings, *until*
it reaches the attractor. At every step in this transformation the Euclidean
distance *d *between the point and the
transformed point is measured. The total distance* D*(*x* *, y*) = *d1 *+ *d2
+ ...+ dn*, where *n *equals
the number of iteration steps needed for a specific point to reach the
attractor. In applying this procedure, to every point in the array a number is
assigned, which is to be interpreted as the convergence distance from a point
to the attractor, following the transformations determined by the random
string. Using this number, a colour can be assigned to each point. When there
is indeed a structuring principle behind the individual orbits of points, we
should expect that when performing this procedure, the colours will not be
distributed at random, but should display discrete transitions, forming
structures. In Figs. 2-4 examples are shown of applying this method to three
different attractors.

* *

*Fig. 2 Application of the
method discussed. The attractor used is the Sierpinski triangle, which is
framed in black here. As can be observed, this original attractor is part of a
greater Sierpinski-like structure. *

Fig. 2 Here the method is
applied to Barnsley’s fern. Again one can observe that the original attractor –
again framed in black – is one leaf of a greater fern.

*Fig. 4: Application of the
method discussed, the attractor having a tree-like structure. This original
attractor is the small, dark little tree which is here again framed. *

As can be concluded from these computer generated images it
is clear that despite the individualness of the orbits, there is clearly an
ordering principle behind these orbits. One can clearly observe that groups of
neighbouring points form clusters, cfr. the discrete transitions in the
colouring due to discrete transitions in convergence distances. Moreover, some
of these clusters resemble the original IFS-fractal, which is framed in black
in Figs.2-4. When looking at the overall structure it even seems to be the case
that this original attractor is part of one greater fractal structure. So one
can conclude that, even if each point has a unique orbit under the
transformations, there is clearly a structuring principle which classifies
these points in fractal clusters. In trying to find an explanation for this
clustering, further experiments have shown that this structuring is caused by
the number of iteration steps to be performed on each point before reaching the
attractor. When plotting only the points with a given speed, the images
immediately show that each structure is indeed formed by points which have the
same number of iteration steps. This is illustrated in Fig. 5 where one such
structure was isolated from Fig. 4, by plotting only those points with speed 2.

*Fig. 5: Visualisation of
only those points that converge to the attractor after the application of the
first two mappings determined by the random string, used in Fig. 4. *

Several
other experiments were performed to further analyse the convergence behaviour
of points when applying the chaos game on them, like for example measuring the
fractal dimension of these structures, but these will not be mentioned here
[see 4]. What is of more importance here is the fact that these experiments
clearly illustrate *how* computer
generated images indeed can be useful instruments for doing research. In the
experiments done here, they gave an immediate answer to a question concerning a
possible structural property of a system.
Once such a question is solved in this way, it is often the case that
the results can be further applied for different purposes. This was also the
case for the results described here.

**3. Applications**

The fact that points in space, which are each transformed
according to the chaos game using the same random string, are fractally
clustered together according to convergence speed, can now be further applied
for other purposes. Two kinds of applications will be discussed here.

**3.1. Graphical Applications**

As could be already noticed in Figs. 2- 4 the procedure
described generates a fractal structure of which the attractor that is used to
generate this fractal is just a small part. In this way the method can thus be
used to expand an attractor. However if one wants to use this as a technique it
is not enough to just perform the operation that was described, since it is
clear from the figures that this expansion seems to go on without bound. This
can be avoided in using the observation that these structures are formed
according to convergence speed. When isolating structures according to speed,
it was observed that the structure formed by points with speed 1 will form an
expansion of the original attractor, points with speed 2 will form an expansion
of the points with speed 1 and thus an expansion of the original attractor,...
Thus to expand the original attractor one just has to add the further
restriction to the algorithm that only those points with speeds smaller than a
chosen value should be plotted.

Figs. 6 The left figure shows the original attractor, while
the picture on the left shows a plot of the original attractor, together with
the points of speed 1. This whole structure is a scaled up reflection of the
original attractor.

In
Fig.6 this was done for the tree-like fractal used in Fig. 4, using a different
random string. The left figure shows the original attractor, while in the
figure on the right the attractor is plotted together with the points with
speed 1. This plot results in a scaled up reflection of the attractor.

As
a further graphical application, which was found through observations made
during the experiments, the possibility of generating contours of the original
attractor should be mentioned. It could be observed in Fig. 5 that when
isolating a structure according to speed, the result indeed resembles the
original attractor, but one part of the plot is less dense then the other part.
This less dense part, is formed by points which are in the direct neighbourhood
of points with speed 1. When looking at the structure formed by points with
speed 1, the same is observed: the plot is divided into a dense and a less dense
part. Here the less dense part is in the direct neighbourhood of the attractor.
Using these observations it is possible to generate the fractal contour of an
attractor. In Fig. 7 the result is shown for the tree-like fractal:

Fig. 7: Plot of
points with speed 1 and 2, that are in the direct neighbourhood of the
attractor.

Concluding,
one can say that in using computer generated images as research instruments it
is possible that as a side-effect of the observations made, other techniques
for generating computer images become apparent.

**3.2. Theoretical Applications**

We
have seen that when analysing the chaos game according to convergence speeds,
Euclidean space is fractally differentiated. One of the further observations
made was the fact that using different random strings, results in very
different visualisations. This procedure is thus sensitive to initial
conditions. In slightly modifying the method these observations can be used to
find typical *visual *representations
of strings belonging to a certain class of strings. The classes of strings that
were investigated are strings that are produced by systems of tags. Tags were
first defined by Emil Post [See 6] and are defined in the following way.
Consider an alphabet A = {*α*1, *α*2,..., *α*n}, over which
an initial string *I* of arbitrary but
finite length is defined, and a
constant natural number *v*. For each letter of the alphabet we then
define a corresponding non-empty string. Depending on the first letter of the
initial word, the respective sequence of letters is then *tagged *at the end of the string, and the first *v* letters are removed. In this way a new string is produced, and
the operation of tagging and removing letters can be repeated. Post discusses
the following very simple example:

*A = {1,
0}; 1 → 1011; 0 → 00; v = 3*

Then choosing for example the
following initial condition “*1011000100010101110*” the following sequence
of strings is produced:

*1011000100010101110*

* 10001000101011101011*

* 010001010111010111011*

* 00101011101011101100*

.............................................

Despite
their seeming simplicity, tag systems are in general undecidable. Practically
this means that given a tag that seems undecidable, its behaviour is unpredictable.
One of the undecidability questions that are related to tags is the following
question: given an arbitrary string, will it ever be produced by a given tag
system. The research done by the author on this subject is not yet completed,
but in trying to approach this problem in a more experimental way, the method
here discussed was used in the following way. Given for example the above
discussed tag system developed by Emil Post, and an arbitrary initial
condition. When visualising the strings according to the above discussed
method, one cannot find any generality in the images determined by these
strings. Each string results in a different picture. This problem can be
overcome as follows. Given the binary permutations of a given length *n*, for example the 32 permutations of
length 5, one can analyse a string according to the yes or no presence of each
of these bit strings in the string. Then a list is made of these bit strings
that are part of the string. Using then for example the IFS-transformations
that result in the tree-like fractal, the transformations to be performed are
now determined in the following way. The first transformation to be performed
is determined by the first bit-string in the list. This bit string is converted
to its decimal value, and then divided by the number of IFS-transformations, in
this case 5. It is then the remainder of this division that determines the
transformation to be performed, since this remainder varies between 0 and 4.
For example, when the bit string 101100 is the sixth bit string of such a list,
the sixth transformation to be performed is then the remainder of the division
of the decimal value 44 of this bit string by 5, which is 4. In this way a
string produced by a tag is thus transformed into a new string consisting of
digits varying between 1 and the number of IFS-transformations. This
modification however is not enough so as to get a *complete* visual representation of a string produced by a specific
tag. This is due to the fact that when looking at the maximum convergence
speeds these are rather low. The maximum speeds in Figs. 2-4 are respectively
15, 9 and 13. This implies that when using this procedure for the transformed
strings, only the first 15, 9 and 13 digits respectively will be represented in
the visualisation of the string. To overcome this problem, two attractors
instead of one can be used. Without going into the details of how this is done,
experiments have shown that we can now have a complete representation of a tag
string, reaching maximum speeds which equal the number of bit strings a tag
string is analysed in. In fig. 8 the
result is shown of applying this procedure for an arbitrary chosen string
produced by the tag discussed by Post, for a random initial condition. The
question then is if this visualisation can indeed be called a typical
visualisation of this tag. This is indeed what seems to be the case.

*Fig. 8: Typical visualisation of a string produced by
Posts’ tag, through the method discussed.
*

** **

The
procedure was tested for this tag for ten different random initial conditions.
For each of these initial conditions, 300000 strings were produced, out of
which 10000 randomly chosen strings were analysed. The results of this analysis
for permutations of length 5, resulted in 0,0237% deviating strings i.e.strings
that dot not result in the visualisation shown in Fig. 8. This means that there is a very small chance
that strings produced by this tag will not result in the visualisation of Fig.
8, so it can indeed be called a *typical*
visualisation of this specific tag. Moreover, when applying this same procedure
to other tags, which were selected through an algorithm which tests for *possible* undecidability, the ‘typical’
visualisations for each of these tags was always different for each tag, so
that this method can be generalised to other tag systems. Without going further
into the details of the research already performed in this context, one can
conclude then that this method is at least a practical answer to one of the
undecidability problems that are related to tags, be it a probabilistic one.
Perhaps further research on this subject can find other methods for tag
recognition. In spite of the unavoidable undecidability, it can then be
possible to increase the probabilities in complementing these methods with the
method here discussed.

**4. Conclusion**

Using
computer generated images as research instruments is indeed one of the
interesting possibilities that became available to scientific research with the
commercialisation of the computer. It is clear that these images *as* images are not enough on their own to
perform research. The right questions have to be asked and one must be able to
perform the right experiments and to interpret these results. Notwithstanding
these remarks, speaking from my own experiences as a non-mathematician, it must
be noticed that, as a technique, it offers the possibility of performing
research in a more intuitive way. Sometimes one can get frustrated for not
knowing the right mathematical formulations for found results, but most of the
time, I was able to answer my questions by performing new visual
experiments.

**References**

** **

[1]
Martin Gardner, *The fantastic
combinations of John Conway's new solitaire game "life” *in: *Scientific American, *223, 1970, 120-123.

[2]
Stephen Wolfram, *Random Sequence
Generation by Cellular Automata* in: *Advances
in Applied Mathematics*, 1986, 123-169.

[3]
Benoît Mandelbrot, *Les Objets Fractals:
Forme, hasard et *dimension, Paris:Flammarion, 1995.

[4] Liesbeth De Mol, *Study of Fractals derived from IFS-fractals
by Metric Procedures*, submitted to *Fractals.*

[5] H.Peitgen, H.Jurgens and D. Saupe,
*Chaos and Fractals. New Frontiers of
Science*.* *Berlin: Springer Verlag,
1992.

[6] Post, E. L. *Formal Reductions of the General Combinatorial Decision Problem.* In
*Amer. J. Math.* 65, 197-215, 1943.