A "still Life" is a subset S of the square lattice Z^2 fixed under the
transition rule of Conway's Game of Life, i.e. a subset satisfying the
following three conditions:
1. No element of Z^2-S has exactly three neighbors in S;
2. Every element of S has at least two neighbors in S;
3. Every element of S has at most three neighbors in S.
Here a ``neighbor'' of any x \in Z^2 is one of the eight lattice points
closest to x other than x itself. The "still-Life conjecture" is the assertion
that a still Life cannot have density greater than 1/2 (a bound easily
attained, for instance by {(x,y): x is even}). We prove this conjecture,
showing that in fact condition 3 alone ensures that S has density at most 1/2.
We then consider variations of the problem such as changing the number of
allowed neighbors or the definition of neighborhoods; using a variety of
methods we find some partial results and many new open problems and
conjectures.Comment: 29 pages, including many figures drawn as LaTeX "pictures
