Vitaliy Kaurov Blog

Another short code I wrote in Mathematica language was accepted by The Wolfram Demonstrations Project. The Demonstration shows the evolution of two elementary cellular automata (CA) sharing several cells. CA are often treated as isolated systems with simple cyclic or Dirichlet boundary conditions. Realistic systems, in contrast, interact with the environment through a boundary. Boundaries can be as simple as solid body surfaces, as complex as walls of living cells, or even have non-geometric nature as the boundaries of social systems. A boundary, having an intricate structure and being a coupling link to the environment, can strongly influence the system dynamics. Here two elementary CA colored red and blue interact via a boundary consisting of black shared cells. The configuration of coupling is schematically shown on the image at the lower-left corner of the graphic. Even the single cell boundary can significantly alter the dynamics of CA. If the black overlap is made significantly large, it can be considered as a 2-color range-2 CA interacting with two elementary CA. The boundary in this case is the line separating CA of different color. The CA exchange information via the bordering cells. The Demonstration also shows how larger neighborhood CA patterns arise from the multiple action of smaller neighborhood CA. Click on the picture below for more technical description and to play with the program. Click on the YouTube icon to view a short video of the demonstration.

The Wolfram Demonstrations Project published a program I wrote in Mathematica code that visualizes recursive formula attributed to Peter de Jong:

It can produce beautiful nontrivial structures called attractors. Click on the picture to go to the Wolfram demonstration page where you can animate and interact with this program.

This is another Mathematica program I wrote published by the Wolfram Demonstrations Project. The program animates flickering fire with a simple mathematical algorithm using Wolfram rule-99 cellular automaton. The algorithm is based on calculation of eigenvalues of a matrix representing the cellular automaton evolution. To play with this program and to read more detailed explanation click on the picture. Click on the YouTube icon to view a short video of the demonstration.

Trivial systems with primitive standalone behavior can produce rich dynamics working in collaboration. The Wolfram Demonstrations Project published my program which allows to couple up to four different one-dimensional elementary cellular automata. It produces some interesting patterns and behaviors beyond those of standalone cellular automata. The program is basically applies idea of coupled recursive maps (see example here) to cellular automata.  Click on the picture below for more technical description and to play with the program.

The Wolfram Demonstrations Project published my program which shows how to reproduce some two-dimensional (2D) cellular automata (CA) with one-dimensional (1D) rules. Thus a large subclass of 2D CA can be conveniently labeled through the standard Wolfram indexing of 1D CA. One step of time evolution of a 2D CA is obtained in two stages. First, apply a 1D rule to all the ROWS of a 2D initial condition. Then to the result apply another 1D rule, but now to all the COLUMNS. This can be mapped onto the subclass of two-color 2D CA with a nine cell square neighborhood. It is useful to devise smaller subclasses of the general case to manage classification of such a large number of CA. Examples of such subclasses are totalistic CA, outer-totalistic CA, and the new subclass demonstrated here. Click on the picture below for more technical description and to play with the program. Click on the YouTube icon to view a short video of the demonstration.