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Mathematics LibreTexts

11: Cellular Automata I - Modeling


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  • 11.1: Definition of Cellular Automata
    “Automaton” (plural: “automata”) is a technical term used in computer science and mathematics for a theoretical machine that changes its internal state based on inputs and its previous state. The state set is usually defined as finite and discrete, which often causes nonlinearity in the system’s dynamics.
  • 11.2: Examples of Simple Binary Cellular Automata Rules
    The two exercises in the previous section were actually examples of CA with a state-transition function called the majority rule (a.k.a. voting rule). In this rule, each cell switches its state to a local majority choice within its neighborhood. This rule is so simple that it can be easily generalized to various settings, such as multi-dimensional space, multiple states, larger neighborhood size, etc. Note that all states are quiescent states in this CA.
  • 11.3: Simulating Cellular Automata
    Despite their capability to represent various complex nonlinear phenomena, CA are relatively easy to implement and simulate because of their discreteness and homogeneity.
  • 11.4: Extensions of Cellular Automata
    So far, we discussed CA models in their most conventional settings. But there are several ways to “break” the modeling conventions, which could make CA more useful and applicable to real-world phenomena. Here are some examples.
  • 11.5: Examples of Biological Cellular Automata Models
    In this final section, I provide more examples of cellular automata models, with a particular emphasis on biological systems. Nearly all biological phenomena involve some kind of spatial extension, such as excitation patterns on neural or muscular tissue, cellular arrangements in an individual organism’s body, and population distribution at ecological levels. If a system has a spatial extension, nonlinear local interactions among its components may cause spontaneous pattern formation, i.e., self-