I presume it is well-known that Brownian motion on Z^n is macroscopically isotropic, and therefore macroscopically indistinguishable from continuous Brownian motion on R^n? Basically, the position of a particle in each dimension at time t follows a binomial distribution, which asymptotically approaches a normal distribution. So, the probability density function in Z^n approaches a Gaussian 'hill' over n dimensions (easiest to visualise when n = 2). If the coordinates are x_1, x_2, x_3, ... , x_n, the probability density function is proportional to: e^-(x_1² + x_2² + x_3² + ... + x_n²) = e^-d² where d is the distance from the origin. Similarly, lattice gases can exhibit macroscopic isotropy, and thus cellular automata should be capable of approximating reaction-diffusion systems to arbitrary accuracy. Sincerely, Adam P. Goucher