6. Conclusion

We have shown that two important features that current discrete models have - crystalline geometry and global synchronization - are unnecessary to produce a model with physical behavior. To compare with what happens in three-dimensional space over time, one needs to put constraints on the interconnections of the lattice and the schedule of the updates to be ``spacelike'' and ``timelike''. However, it is not necessary for any one site to exploit detailed knowledge about what time other sites update, and about their precise spatial arrangement.

In particular, synchronization of all sites in space with each other is not required - either actual synchronization, or schemes that effect synchronization in an asynchronous environment. Also, we found that anisotropies due to the irregular lattice can be compensated for by using a higher density of sites, so these will probably not be significant at large scales. Synchrony and crystalline geometry, then, are useful when we want to simulate the models on machines that have these properties, but are not intimately connected with discrete physical models. The wave models presented do without both, but agree well quantitatively with partial differential equations.

Many interesting questions remain. One topic for further investigation is to determine what other kinds of physical systems, including nonlinear ones, can be modelled using this approach. Also, in the model presented, the states of the sites in the wave model are continuous quantities; it is open whether a wave model with the same restrictions, but where sites can take on a very restricted set of states (such as 0 or 1), is possible, and if not, what extra conditions would be needed.

We expect that, if amorphous computers become feasible, models like the class presented will become of practical interest.