Resonances in 2D CA of Class IV

Else Nygren

Uppsala University

In a famous paper of 1984, Packard and Wolfram describe the properties of two-dimensional cellular automata, 2D CA. The authors describe the behavior of some 2D CA of class IV. This class of CA shows interesting behavior with a mixture of chaotic, static, and periodic ingredients. One famous such rule is the Game of Life, which has been extensively described in the literature.

I have investigated a family of 2D CAs that all exhibits the behavior of class IV. In particular I have investigated what happens when such a CA evolves from small seeds in restricted spaces. It was found that for most seeds a growing structure evolves, but as soon as this encounters the border of the CA universe, the symmetry is broken and the CA settles in a state consisting of distributed static or periodic small structures. However, there seems to exist a critical size where the growth will result in a dynamical resonance. Such a resonance consists of a single periodically changing structure often with a high degree of symmetry, with some static cells and others dynamically changing. We call this a resonance since it only appears in certain combinations of seed and universe size. We can assess a measure of the “energy” of such a resonance in terms of how many cells that change on average per CA time step. In the evolution of such a CA from the initial seed, several temporal but unstable resonances are formed. We can often see distinct phases of successive resonances of lower and lower energy. Each time a temporal resonance emits gliders (photons), its energy is reduced until it finally settles down into a resonance pattern of the lowest possible energy.

There is a large number of possible initial seeds of size 2 x 2 cells. Many of these are redundant, and we can find that there is actually only a limited set of types of such seeds that gives characteristically different resonances. Type 1 shows a resonance that is symmetrical with respect to rotation as well as mirror reflection in the x- and y-planes. Type 2 is likewise symmetrical but has a hole in the middle. Type 3 is also symmetrical except for a small area around the center that is only diagonally symmetric. Type 4 is diagonally symmetrical but not rotationally or in terms of mirror reflection. It also has a center area that shows a different symmetry. Type 5 is interesting, it has the properties of type 2 but has a small particle in the center hole that is slowly rotating around the inside. You could say that this resonance possesses some kind of momentum. Type 6 shows symmetrical properties except for the center area where there is a periodic right to left oscillatory movement.

The notion of having a confined space for the CA to evolve seems very constraining. There seems to be a need for some “resistance” in order to develop the resonance. What if instead of encountering a border the growing CA would encounter another growing CA? It turns out that this indeed works to produce resonances. Some CA seeds make up nice grids together in this way. Others interfere destructively with each other no matter the separation between the seeds. But with this notion of seeds making up constrained spaces for each other it is possible to build grids of a whole variety. Most specificially the kind of grids normally used in material sciences.

Now, having the possibility to construct grids, we can study what happens when we emit “light” in the forms of rays of photons against such grids. We can see that depending on the phase relation between the photon and the resonances in the grid the photon is either absorbed, scattered with a change of wavelength or reflected back from the material surface.

In summary, resonances in 2D CA of Class IV seem to exhibit many interesting “physics-like” behavior. We can see discrete energy states, excitation, emission of particles, reflection, asorbtion, and scattering phenomena. We can also see that a photon can be scattered by “virtual particles” interpreted as temporal resonances. We can also see some quantum-like phenomena as the particular outcome of an interaction that seems random as seen from a distance. The resonances also show an ability to self-synchronize. If they start out being out of phase, they spontaneously synchronize with each other, the result being a combined resonance completely phase-tuned.

[presentation materials]

Created by Mathematica  (May 23, 2006)