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- W1890114378 abstract "A 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." @default.
- W1890114378 created "2016-06-24" @default.
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- W1890114378 date "1998-01-01" @default.
- W1890114378 modified "2023-09-23" @default.
- W1890114378 title "The Still-Life Density Problem and its Generalizations" @default.
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