Mathematics – Combinatorics
Scientific paper
1999-05-31
Pages 228--253 of_Voronoi's Impact on Modern Science, Book I_ (P. Engel, H. Syta, eds.; Institute of Math., Kyiv 1998 = Vol.21
Mathematics
Combinatorics
29 pages, including many figures drawn as LaTeX "pictures"
Scientific paper
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.
No associations
LandOfFree
The still-Life density problem and its generalizations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The still-Life density problem and its generalizations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The still-Life density problem and its generalizations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-596238