Experimental Mathematics [2]

An increasingly important application of computers is in doing experiments on mathematical systems. Using a computer, one can find out how a mathematical system behaves, even though with conventional mathematical techniques one cannot carry out a complete analysis. (Nevertheless, having found evidence that a certain mathematical fact is true, it may be possible to prove it using conventional mathematical techniques.)

Probably the most significant consequence of the experimental mathematics method is that it is making possible the investigation of many complex systems hitherto inaccessible to theoretical study. It is central to such fields as dynamical systems theory.

As one example of experimental mathematics, I discuss a class of systems called cellular automata that I have studied extensively. These are mathematical systems whose microscopic construction is very simple, yet whose overall behaviour can be highly complex. They probably capture the essential mathematical mechanisms by which complexity is generated in many natural systems.

[ Figure 3 ] Patterns generated by some simple cellular automata (with two possible values for each site, and nearest-neighbour rules).

In the simplest case, a cellular automaton consists of a line of sites, each with say two possible values. The values are updated in a sequence of discrete time steps according to a fixed rule that depends on neighbouring values. Figure 3 shows some examples of patterns produced in this way. Considerable complexity is evident. But a number of definite features, such as self similarity, are seen. Almost all of these features were discovered using experimental mathematics, as a result of explicit computer simulations.

Cellular automata provide models for many physical, biological and other systems. They can be considered for example as approximations to partial differential equations. Not only is a discrete lattice taken in space and time, as in standard numerical analysis, but in addition discrete values are assumed for the variables at each site. Just one or two bits of information are included at each site, rather than the 32 or 64 bit numbers commonly used in the standard numerical analysis approach on digital computers. Local averages must be performed to find continuum quantities such as fluid density. The cellular automaton approach to a problem like fluid flow can be viewed as intermediate between molecular dynamics, in which very many molecules must always be averaged over to obtain a result, and the continuum partial differential equation approach, which gives directly specific results for continuous parameters.

An important advantage of cellular automaton models is that they can be implemented very efficiently by the forthcoming generation of massively parallel computers. The architecture of such computers is close to the architecture of cellular automata, and apparently also to the ``architecture'' of many physical systems.

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