CA3D and
CATree: Creating virtual objects by using Cellular Automata.
Prof. E. Bilotta.
Department of
Linguistics, University of Calabria, Arcavacata di Rende (CS), Italy.
email:
bilotta@unical.it
Dr. A. R. Gabriele.
Department of
Linguistics, University of Calabria, Arcavacata di Rende (CS), Italy.
email:
a.gabriele@unical.it
Dr. M. G. Lorenzi.
Department of Linguistics, University of Calabria,
Arcavacata di Rende (CS), Italy.
email:
m.lorenzi@unical.it
Prof. P. Pantano.
Department of
Mathematics, University of Calabria, Arcavacata di Rende (CS), Italy.
email:
piepa@unical.it
Abstract
In
the following work we present artistical
forms, realized using two software applications that we designed and
implemented ad hoc, CA3D and CATree.
The
first tool produces fractal structures. Simulating the CA evolution, this
software makes threedimensional patterns, by representing each cell of the CA
by a cube of a different colour, depending on its state and on its ruletable.
We have called these structures "Fractal Solids".
CATree
generates fractal structures utilizing cylindrical forms to represent the
cells. In addition, it allows to modify the dimension of the cylinders during
the CA evolution. In fact, while in the visualization with cubes their
dimensions remain constant, in the visualization with cylinders height and
radius vary during the simulation. We have called these structures
"Fractrees", considering the characteristic configuration, similar to
trees, they show.
Fractal solids and Fractrees can be used in different ways. For example,
in virtual worlds and environments, they can represent the objects their form
reminds of (pyramids, trees, etc.) or
they can be used as units to build other forms (cities, animals, etc.). Some
experimentation we have been carried out include all these representations,
used to create characters, pictures, stories and videos.
Cellular automata (CA) were
introduced by von Neumann and Ulam as simple models in order to study some
biological processes. A cellular automata is a discrete dynamical system, that
is, a system in which the variables that determine and describe their evolution
(space, time and states) assume discrete values.
Selfreplication,
the capacity of the cellular automata to evolve reproducing some patterns, is a
phenomenon that is extensively investigated [14]. In particular, we studied
twodimensional ktotalistic cellular automata [3] that present the
selfreplication phenomenon [35]. In order to investigate comprehensively the
obtained spatiotemporal patterns, we implemented the CA3D software, which
simulates the evolution of ktotalistic twodimensional cellular automata. The
objects obtained by the simulations present different characteristics, typical
of fractal objects [68], such as selfsimilarity and fractal
dimension.
These
two characteristics are easily identifiable in such objects. The first one is
immediately visible, while the fractal dimension has been calculated using the
boxcounting [68] method.
The beauty of these objects and their characteristic forms let us create
virtual objects and worlds [911].
2. Emerging forms in CA3D
CA3D
allows the representation of the evolution of a cellular automata in the
spatialtemporal dimension.
The evolution of the automata in
time can be represented into the threedimensional space, adding a third
dimension to the cell of the plane, for example using a cube for each cell.
The following figure reports a cellular automaton in its initial
configuration (Figure 1.a) and the first steps of its evolution (Figure 1.b).
a 
b 
Figure 1. : A cellular automaton in its initial
configuration (a) and the first six steps of its evolution (b).
The
cells are represented as cubes in the space, and positioning the various
configurations one above the other, from top to bottom, we obtain the
spatiotemporal evolution of the cellular automata.
The next example (Figures 34) shows the evolution in space of the
initial configuration represented by following the matrix (Figure 2):
a 
b 
Figure 2. : Initial configuration of the cellular
automaton (a) and the matrix it represents (b).
The automata evolves for 31 temporal steps, obtaining the model shown in
Figures 34.
Figure 3: Three
dimensional view of the obtained object 
Figure 4: a) and b) show two lateral views
of the object; c) and d) report two different views: from below c), and from
above d). 
The object we obtained have been called Fractal Solids, since they
exhibit selfsimilarity characteristics (Figure 5).
Figure 5:
Selfsimilarity of a Fractal Solid, from a lateral view and in perspective.
CA3D implements a procedure that calculates the dimension of each object
that has been evolved using the boxcounting method, that in our case is a
fractal dimension.
Figure 6: Examples of
Fractal Solids. The fractal dimension of the objects is the following:
a=2.7215; b=2.4207; c=2.5042; d=2.9302; e =2.8277; f
=2.6684.
Subsequently, a
new tool, called CATree, has been implemented, that allows the generation of
new fractal structures which make use of different geometrical forms to
represent the cells (cylinders rather than cubes). In addition, we decided to
modify the dimension of the cylinders during the CA evolution. In fact, while
in the visualization with cubes thier dimension remains constant, in the
visualization with cylinders height and radius vary during the time steps of
the simulation. These structures have been called Fractrees, considering the
characteristic configuration they show if we rotate their position upside down.
Some examples are shown in Figure 7.
Figure 7: Examples of Fractrees.
In Figure 8 two
trees obtained introducing a factor of probability to decide the creation of a
branch.


Figure 8: Two examples of trees evolved taking into consideration a
factor of probability.
Artists have
been extensively using, during the last years, new methods and instruments,
inspired by mathematical models, to generate artistical and creative forms, as
well as artificial worlds, through which new experimentations can be carried
out and cultural products of high expressiveness can be created. ESG group creates
images, sounds and videos using mathematical models related to complexity,
chaos and artificial life, among the others [910]. In addition some
experiments of storytelling have been carried out: one of it represents the
evolution of a robot and the exploration of artificial worlds, made by
different configurations of one and twodimensional cellular automata. The
sound track was also created starting from one and twodimensional cellular
automata. [11]
In the poster (Figure 9)
characters, landscapes and other objects are realized using one and
twodimensional cellular automata. In particular, a city landscape has been
created, using gliders as buildings and virtual trees, created respectively
using CA3D and CATree (bottom). All the elements of the composition (Figure
10) are then illustrated separately, showing the one or twodimensional CA the
are generated from (center). From the top, the evolution steps of a Fractal
Solid, created using CA3D, fades into the generated form; the same form is visualized,
upside down, with CATree. Scientific references, which are at the basis of
these results, are illustrated on the sides. All the poster is generated
following a metaphor: the evolution of the forms in time steps, and their
subsequent superimposition, to arrive to the complete (flattened) composition.
Some characteristics of the mathematical models that have been used are present
in its logic of composition: for example, the same object is present at
different scales in different points. Finally, all the elements follow the
graphical layout of the Fractree on the background.
Figure 9: Poster
Figure 10: Example of artistical composition
References
[1]von Neumann J. (1966). Theory of SelfReproducing Automata, Burks A.W Ed. University of Illinois
Press, Urbana.
[2]Wolfram S. (2002). A New Kind of Science, Wolfram
Media, Inc.
[3]Bilotta E., Lafusa A., Pantano P., (2002). Is
selfreplication an embedded characteristic of artificial/living matter?, In:
Artificial Life VIII: Standish, R.K., Bedau, M.A., Abbass, H.A., Eds.; The MIT
Press: Cambridge, MA; pp. 38–48
[4]Bilotta E., Lafusa A., Pantano P. (2003).
Lifelike selfreproducers, Complexity, 91, pp. 3855
[5]Bilotta E., Lafusa A., Pantano P., (2003).
Searching for complex CA rules with GAs, Complexity, 83,5667.
[6]B.B. Mandelbrot, (1987). Gli oggetti frattali: forma,
caso e dimensione, Einaudi,Torino.
[7]Barnsley M.F. (1988) Fractals Everywhere, Academic
Press, New York.
[8]Peitgen H.O., Jurgens H., Saupe D., (1992). Fractals
for the classroom, SpringerVerlag, New York.
[9]Bertacchini P. A., Bilotta E., Pantano, P., Di Bianco
E., Fiorelli D., Gabriele A., Gervasi S., Lorenzi M., Sposato F.,
Vena S. (2003) Report sulle attività artistiche dell'Evolutionary Systems
Group. I° workshop Italiano di Vita Artificiale, Università della Calabria,
Settembre 2003.
[10]E. Bilotta, M. Lorenzi, P. Pantano, A. Talarico
(2004). Poster: Art inspired by cellular automata. NKS Conference, Boston April
2225, 2004.
[11]P. A. Bertacchini, M. Lorenzi, A. Talarico (2004)
Storytelling: creating artificial worlds using mathematical models. Bilotta E.,
Francaviglia M., Pantano P. (a cura di) Applicazioni della Matematica
all’Industria Culturale, Atti del Minisimposio, Convegno SIMAI 2004, CDROM,
Venezia, 2324 settembre 2004.