The science of cellular automata is still quite young, and we're not yet surrounded by everyday applications of it. However, as Rudy Rucker says in his book Seek!, "[I]n coming years you won't be able to watch television for an hour without seeing some kind of CA [in the form of] commercial computer graphics."
Some visual representations of CA, such as this color diagram of Rule 110, are quite beautiful:
Figure 1. Color representation of Rule 110
You'll find more examples in Stephen Wolfram's book A New Kind of Science or at his Web site.
CAs also have possible applications in technology. As Wolfram explains in his book, we often invent quite complicated mechanisms to construct technology. On the other hand, CAs are capable of achieving quite complex behavior even though they are defined by very simple rules. It might be possible to use techniques based on CAs to solve technological problems in new ways that make our current solutions look like Rube Goldberg-style contraptions.
Wolfram also points out that we now explain physics and the other sciences mainly by the use of equations. However, simple programs, such as CAs, can produce quite complex behaviors. It might be possible to solve scientific problems that seem intractable using an equations-based approach, by thinking about these problems in terms of computations or programs.
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