The stillLife density problem and its generalizations
Abstract
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^2S 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 "stillLife 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1999
 arXiv:
 arXiv:math/9905194
 Bibcode:
 1999math......5194E
 Keywords:

 Mathematics  Combinatorics;
 05B40;
 05B45
 EPrint:
 29 pages, including many figures drawn as LaTeX "pictures"