© Copyright 1993 Mario Hilgemeier
We see that a string can be the description of the state of a system. If the system is a plant, then L-systems may be the appropriate string rewriting systems (SRS) to model development patterns. Other dynamical systems, in celestial mechanics for instance, are described by a set of numbers (three dimensional coordinates and impulses) and the equations governing the relations of the bodies (gravitation). The next set of numbers is derived from the previous by iteration because gravitational systems of more than two celestial bodies are not exactly solvable by symbolic differential equations. If a state is characterized by n numbers, it can be represented as a point in n-dimensional space, the so-called phase space. As the system evolves, the point hops through phase space, not always very smoothly (large jumps happen). Although the description of the dynamical system by differential equations yields continuous movement, for reasons of computability the representation is transformed into a discrete dynamical system where iteration is used. But the introduction of iteration leads to discrete "jumps" of the system state (quantization). Watching only one coordinate of the phase space is like measuring any parameter (e.g. the rainfall per day) of a dynamical system (say, the weather). What you get is a time-series, sometimes a fractal curve (this depends on the type of dynamical system). If you enlarge portions of a fractal curve, they look "like the whole curve;" you find the same patterns after each renewed magnification. Any iteration of the GZR can be seen as a time-series. Figure 8 shows an example for the 49th iteration. How I arrived at this curve is shown in pseudocode 3. This curve approaches self-similarity because of the iteration rule of the GZR. But the self- similarity is impure, because the line segments are finite.
Figure (7.)8 When you examine the phase space for a given dynamical system, you find that certain regions will never be visited by the moving system state. In other regions however, the point density is very high - these are preferred states, visited often by the system. Phase spaces with two or three dimensions can be represented graphically very easily. Plot a point for each iteration of a nonlinear dynamical system. The geometrical shapes that evolve may be peculiarly beautiful or fractal [8, 9]. If the shapes are fractal, they are called strange attractors. Many readers may be familiar with the beautiful fractal images from the theory of dynamic systems[11, 14]. The length of time for which predictions are valid for nonlinear dynamical systems (weather, solar system) depends on our (limited) knowledge of the initial conditions. Because system states that are very close in phase space can get farther and farther apart in the future, our prediction ability is as limited as our knowledge of the initial conditions. Systems of orbiting bodies in celestial mechanics that have rational proportions of orbital periods become unstable sooner or later. Therefore it has been said that the stability of the solar system crucially depends on the irrationality of the proportions of the orbital periods. Well, for the foreseeable future, no disaster looms, as far as computer models go today. Computers often model states of the real world. By computation, limited prediction and explanation of the phenomena of the world becomes possible.
A robot must have an internal model of the world around it - at least of those qualities of the environment that it needs for its function. For example, a washing machine is concerned with laundry type, water, temperature, detergent, time and the like. The model of the environment of the robot is represented in the tiny computer by a set of numbers and rules, hence we have met another kind of phase space here. The robot's state has a position in this many-dimensional phase space. The robot's internal representation of its "position in the world" moves in phase space. If it is to be a successful robot, it will learn what the best actions are in certain regions of the phase space (e.g. "if the is water too cold for linen, start heater"). Of course, if the robot is to learn by trial-and-error, it must somehow evaluate which actions were successful. We've entered the world of artificial intelligence (AI). AI is concerned with reasoning, learning and making analogies in new situations. The kinds of logic used are not always of the simple yes and no type, and learning paradigms abound. Sadly, I can't go further into this for now. If you want to read more, see the references [15, 16]. The point I wanted to make here is that fractals may have something to do with (machine) intelligence. Iterated systems (IS), such as computers, can show chaotic (fractal) behavior. Many questions arise. Here are few: Does the robot's state in phase space move toward a strange attractor? What are the consequences? For what purpose can we use the dynamics of learning systems if they are chaotic? Is there a danger in the low predictability of chaotic systems? Engineers have just begun to exploit nonlinear chaotic systems. Some chaotic systems can be controlled by a chaotic signal and forced to express the desired behavior . Can chaotic systems be used to generate "creativity" of robot artists? Or even for the automatic generation of unique stories, films, architectural designs, virtual reality scenarios or other cultural artifacts? In Birkhoff's aesthetic theory, art is defined as something not boring, but at the same time not too surprising . Fractals surely fit this simple criterion. So they may be useful for everyday design (and - in the hand of humans - for works of art).next page content page
© Copyright 1993, Mario Hilgemeier, email: contact