You can regard many cellular automata as discrete approximations
to particular differential equations. The best known perhaps is
the repeated replacing of a center cell by the average of its
neighbors as a way of solving the Laplace equation. What has
happened in this case is that the process has been reversed and
the discrete approximation has been taken back to the continuous
differential equation. Now the cell becomes a circular disk and
the nearest neighbors form a ring around it. The rules now
concern integrals taken over the relevant areas and the rules
become a function which takes the integrals to new values of the
cell. Now we have a new video by Tim Hutton of SmoothLifeL (a
particular version of the rule) in action complete with gliders:
"SmoothLife is a family of rules created by Stephan Rafler. It
was designed as a continuous version of Conway's Game of Life -
using floating point values instead of integers. This rule is
SmoothLifeL which supports many interesting phenomena such as
gliders that ca